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+Inertial to viscous coalescence of liquid lenses: a lattice Boltzmann investigation
+Thomas Scheel,1, 2, ∗ Qingguang Xie,1 Marcello Sega,3, 1, † and Jens Harting4, 5, ‡
+1Helmholtz Institute Erlangen-N¨urnberg for Renewable Energy (IEK-11),
+Forschungszentrum J¨ulich, Cauerstr. 1, D-91058 Erlangen, Germany
+2Department of Physics, Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg, Cauerstr. 1, D-91058 Erlangen, Germany
+3Department of Chemical Engineering, University College London, London WC1E 7JE, United Kingdom
+4Helmholtz Institute Erlangen-N¨urnberg for Renewable Energy (IEK-11),
+Forschungszentrum J¨ulich, Cauerstr. 1, D-91058 Erlangen, Germany
+5Department of Chemical and Biological Engineering and Department of Physics,
+Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg, Cauerstr. 1, D-91058 Erlangen, Germany
+(Dated: January 16, 2023)
+Liquid lens coalescence is an important mechanism involved in many industrial and scientific ap-
+plications. It has been investigated both theoretically and experimentally, yet it is numerically very
+challenging to obtain consistent results over the wide ranges of surface tension and viscosity values
+that are necessary to capture the asymptotic temporal behavior in the viscous and inertial limits.
+We report results of massively parallel simulations based on the color gradient lattice Boltzmann
+method, which overcome these limitations, and investigate the scaling laws of both regimes. For the
+two-dimensional case we find good agreement with the similarity solution of the thin-sheet equation,
+where in the viscous regime the connecting bridge grows linearly with time and in the inertial regime
+proportionally to t2/3. In three dimensions, the viscous growth of the bridge also exhibits a linear
+time dependence, while in the inertial regime the growth of both the bridge height and the bridge
+width is proportional to t1/2.
+I.
+INTRODUCTION
+From the formation of raindrops [1] to biomolecular condensates during liquid-liquid phase separation [2], drop
+coalescence plays an important role in many natural phenomena, but finds also broad industrial applications. The
+latter include, for instance, sintering [3, 4], filtration [5, 6], and ink-jet printing of a variety of materials [7–9] ranging
+from solar cells [10–13] to bioengineered tissues [14, 15] and cells [16]. Future improvements in these technologies rely
+strongly on the ability to advance the understanding of the wetting behavior of droplets on liquid substrates as well
+as an accurate knowledge of the interaction between the liquid phases and the dynamics of their coalescence [17, 18].
+Previous research has primarily been focused on the coalescence of freely suspended droplets [4, 19–24] and droplets
+on solid substrates [25–30], while droplets on liquid substrates [31–33] have received less attention.
+The theoretical analysis of the coalescence of liquid lenses, i.e. droplets attached to a fluid-fluid interface, has
+identified two distinct dynamic regimes, which depend on the relative importance of viscous and inertial forces [20].
+Immediately after two droplets get in contact, inertial forces are still small compared to viscous ones, and the con-
+necting meniscus height h0(t) is reported to grow proportionally to the elapsed time t [21, 29, 31, 32]. This linear
+dependence marks the so-called viscous regime. At longer times (or for larger surface tension to viscosity ratios), coa-
+lescence enters the inertial regime where viscous forces become negligible, and h0(t) is reported to grow like h0(t) ∼ t2/3
+for low contact angles θ ≪ 90◦ [29, 32, 34]. The case of contact angles close to 90◦ (as encountered in freely suspended
+droplets) turned out to be a particular one [29], where a scaling h0(t) ∼ t1/2 is often reported [19–21, 29, 31, 35].
+From an experimental point of view it is very challenging to resolve the coalescence process in sufficient detail at
+least for low viscosity liquids such as water. Here, it is practically impossible to observe the viscous regime, because
+inertial effects become dominant at the scale of hc ≈ 15 nm and for times larger than tc ≈ 10−10 s [20]. On the
+other hand, analytical approaches rely on assumptions such as a reduced dimensionality (thin-sheet equation) or
+infinite bridge growth. The difficulties involved in the experimental measurements and the approximations used in
+the analytical treatments have prevented an unambiguous understanding of the scaling laws of three-dimensional
+liquid lenses with arbitrary wetting properties.
+In order to describe the growth dynamics of top-down symmetric liquid lenses (see Fig. 1) it has to be taken into
+account that they involve two principal radii of curvature. As a result, the bridge is characterized not only by its
+height h0, but also by its width w0. Heuristically, one can imagine the evolution of the bridge width w0 in the same
+∗ [email protected]
+† [email protected]
+‡ [email protected]
+arXiv:2301.05498v1  [physics.flu-dyn]  13 Jan 2023
+
+2
+Figure 1. Simulation snapshots of 3D liquid lens coalescence in the y − z plane (top row: side-view) and in the y − x plane
+(bottom row: top-view) at different simulation times t with contact angle θ, lens height h(y, t), minimal bridge height h0 and
+minimal bridge width w0. Lengths and times are provided in lattice Boltzmann units ∆x and ∆t.
+terms as that of its height h0. If the problem was perfectly decoupled into independently evolving height and width,
+one might expect inertial growth rates h0 ∼ t2/3 and w0 ∼ t1/2, the latter because in the y − z-plane projection the
+droplets form initially an angle of 90◦.
+In reality, however, a complex coupling between the two directions is to be expected, which is difficult to model
+with analytical approaches. The impact of their mutual influence on the bridge growth dynamics is an open question
+and one of the main topics addressed in this investigation.
+Computer simulations are in principle a formidable tool to overcome experimental and analytical limitations, but
+accessing both regimes is not an easy task due to the wide range of surface tension and viscosity required [36–38].
+Furthermore, due to the intrinsic multiscale nature of coalescence, one has to resolve orders of magnitude in length
+scales to describe the system from the small initial bridge height to the full droplet size and beyond, including the
+surrounding hydrodynamic flow field [39, 40].
+In this article, we investigate the coalescence dynamics of liquid lenses using the color gradient lattice Boltzmann
+simulation method [41–44]. This method overcomes some of the limitations of the pseudopotential lattice Boltzmann
+approach of Shan and Chen used in previous works [36, 38, 45–47], which was not able to attain the viscous regime.
+The color gradient method allows us to cover both regimes by spanning more than four orders of magnitude in surface
+tension and more than two orders of magnitude in viscosity.
+The remainder of the paper is organized as follows. In section II we introduce the color gradient lattice Boltzmann
+method, while section III summarizes our simulations of two coalescing top-down symmetric liquid lenses. The final
+section provides conclusions and a short outlook on future work.
+II.
+LATTICE BOLTZMANN COLOR GRADIENT METHOD
+Our simulations are conducted with the lattice Boltzmann method on a three-dimensional lattice with 19 discrete
+velocities (D3Q19) [48]. The evolution of the discrete distribution function f k
+i (⃗x, t) for each fluid component k is
+described by the lattice Boltzmann equation
+f k
+i (⃗x + ⃗ci∆t, t + ∆t) = f k
+i (⃗x, t) + Ωk
+i (⃗x, t),
+(1)
+where Ωk
+i is the collision operator, i = 1, ..., 19 specifies the lattice direction and k ∈ {1, 2, 3} the fluid component.
+In the following, we set the time step ∆t = 1 and the lattice constant ∆x = 1 for the sake of clarity without loss of
+generality. The fluid density ρk is obtained from the zeroth moment of the distribution function
+ρk(⃗x, t) =
+�
+i
+f k
+i (⃗x, t),
+(2)
+
+3
+and (in absence of external forces) the macroscopic fluid velocity ⃗uk(⃗x, t) from the first moment of the distribution
+function
+⃗uk(⃗x, t) =
+�
+i f k
+i (⃗x, t)⃗ci
+ρk(⃗x, t)
+.
+(3)
+To model phase separation we employ the color gradient method (CG) which introduces a coupling between the fluid
+components and performs the phase separation in three steps: first, the color gradient, i.e. the direction of steepest
+increase in the density of the respective fluid component, is calculated
+⃗F k(⃗x, t) = ∇
+�ρζ(⃗x, t) − ρξ(⃗x, t)
+ρζ(⃗x, t) + ρξ(⃗x, t)
+�
+,
+(4)
+where ζ, ξ ∈ {1, 2, 3} and ζ > ξ.
+In the next step, also known as perturbation step, the populations that are collinear to the gradient of the color
+field are increased, while those perpendicular to it are decreased, resulting in the appearance of a surface tension term:
+�
+Ωk
+i
+�pert f k
+i (⃗x, t) = f k
+i (⃗x, t) + Ak
+2 |⃗F k(⃗x, t)|
+�
+wi cos2(φk
+i ) − Bi
+�
+.
+(5)
+Here, wi are the lattice weights
+wi =
+�
+�
+�
+1/3
+i = 1
+1/18 i = 2, ... , 7
+1/36 i = 8, ... , 19
+(6)
+and φk
+i is the angle between the color gradient ⃗F k and the lattice direction ⃗ci. Ak is a free parameter determining
+the surface tension and Bi is chosen as to ensure mass conservation:
+Bi =
+�
+�
+�
+−2/9 i = 1
+1/54
+i = 2, ... , 7
+1/27
+i = 8, ... , 19
+(7)
+Finally, the recoloring step separates two phases by distributing the two components to opposite directions
+�
+Ωζ
+i
+�recol
+fi(⃗x, t) = ρζ
+ρ fi(⃗x, t) + β ρζρξ
+ρ2
+cos(φi)
+�
+k=ζ,ξ
+f k,eq
+i
+(⃗x, t)(ρk, 0),
+(8)
+where β is a free parameter controlling the interface thickness (β = 0.99 in all our simulations), fi = �
+k f k
+i and f k,eq
+i
+is the local equilibrium distribution derived from a Taylor expansion of the Maxwell-Boltzmann distribution to the
+second order
+f k,eq
+i
+(⃗x, t) = ρk
+�
+φk
+i + wi
+�⃗ci · ⃗u
+c2s
++ (⃗ci · ⃗u)2
+2c4s
+− ⃗u2
+2c2s
+��
+,
+(9)
+with cs being the lattice speed of sound. The total collision operator of the CG method Ωk
+i is an extension of the
+standard Bhatnagar-Gross-Krook (BGK) collision operator [49]
+�
+Ωk
+i
+�BGK f k
+i (⃗x, t) = f k
+i (⃗x, t) − ωk
+�
+f k
+i (⃗x, t) − f k,eq
+i
+(⃗x, t)
+�
+,
+(10)
+which relaxes the population f k
+i to its local equilibrium with a relaxation rate ωk = 1/τk. From the Chapman-Enskog
+expansion to second order one can derive the relation between relaxation time τk and kinematic viscosity νk of fluid
+k as
+νk = c2
+s
+�
+τk − 1
+2
+�
+.
+(11)
+Finally, the BGK operator is extended by the perturbation and recoloring operators to yield the CG collision operator
+Ωk
+i ,
+Ωk
+i =
+�
+Ωk
+i
+�recol ◦
+�
+Ωk
+i
+�pert ◦
+�
+Ωk
+i
+�BGK ,
+(12)
+which applies in a chain the BGK, perturbation and recoloring operators, in this order, and conserves all collisional
+invariants like mass and total momentum for each fluid component.
+
+4
+III.
+LIQUID LENS COALESCENCE
+Our study focuses on the coalescence of two identical, top-down symmetric liquid lenses. We begin our investiga-
+tion with the quasi two-dimensional case (cylindrical symmetry), before later turning to the fully three-dimensional
+simulations. The droplets are initialized side by side and connected via a contact point (resolved by approximately
+5 lattice nodes). Over time, surface tension drives the interface to minimize the surface area, and a bridge develops,
+which grows until the two droplets have merged into a single larger one.
+The dynamics of the coalescence process is determined by the initial geometry of the droplets [29] and the combined
+effect of inertia, surface tension σ, and dynamic viscosity µ = ρν, where ν is the kinematic viscosity. These quantities
+determine a characteristic velocity scale, also known as capillary velocity, given by the ratio vc = σ/µ. The Reynolds
+number of the coalescing droplets can thus be expressed as Re = ρ σ h0/µ2 [4]. At early times the system is dominated
+by viscous forces, since the bridge height h0 is much smaller than the viscous characteristic length lv = µ2/(σρ) [28].
+In this regime Re ≪ 1 and the flow is described by the Stokes equation. The crossover between the viscous and
+inertial regime occurs at Re ≈ 1. From then on, viscous dissipation becomes increasingly negligible and the dynamics
+of the system is determined by inertial forces.
+For small contact angles, the drop height is much smaller than its lateral extension which allows to apply the
+lubrication approximation, under which the Navier-Stokes equations simplify to yield the thin-sheet equation [50]
+ht + (uh)y =
+0
+(13)
+ρ(ut + uuy) =σ hyyy + 4µ (uyh)y
+h
+.
+(14)
+By solving the thin-sheet equation with the similarity ansatz
+h(y, t) = ktαU(ξ),
+u(y, t) = αk
+θ tβ,
+ξ = θy
+ktα ,
+(15)
+it has been shown that the growth of the bridge between two coalescing lenses exhibits a power-law behavior with
+two asymptotic regimes [32]. In the viscous regime, where viscous forces dominate inertial forces (ρ ≈ 0), the bridge
+height grows linearly in time, h0(t) ∼ t, whereas in the inertial limit h0(t) ∼ t2/3. The two asymptotic regimes as well
+as the crossover region can be described by the universal curve
+h0/hc =
+� 1
+t/tc
++
+1
+(t/tc)n
+�−1
+,
+(16)
+where, in this case, n = 2/3 and tc and hc are the crossover time and height that provide a universal scaling law [32].
+The large viscosity and surface tension range required to reach the viscous as well as inertial regime is a major
+challenge for numerical approaches [51]. So far, the viscous regime was not amenable to the very popular pseudopo-
+tential lattice Boltzmann method of Shan and Chen due to its numerical instabilities at low values of the surface
+tensions [38]. The color gradient lattice Boltzmann method, on the contrary, is stable over a much wider range of
+surface tension values [36].
+For the quasi two-dimensional case we perform simulations of a domain consisting of 4 × 2048 × 768 lattice points
+in x,y and z direction (pseudo 2d) with periodic boundary conditions. The droplets are initialized with a radius of
+282 lattice nodes and a contact angle θ = 30◦. The distance of the droplet edges to the periodic domain boundaries
+are chosen sufficiently large such that their mutual influence across the periodic boundaries can be neglected. Each
+lens was previously equilibrated separately in its surrounding fluid, making sure that the lenses are initially at rest
+and have no initial velocity of approach. To be able to compare our simulation results to the similarity solution of
+thin-sheet theory we ensured that the coalescence process is dominated by the flow inside the liquid lenses by choosing
+the fluid viscosity of the outside fluid to be at least one order of magnitude smaller than that of the lenses.
+We performed a series of simulations by varying the droplet viscosity and surface tension over several orders of
+magnitude to yield low and high capillary velocities, respectively, which allow us to investigate the viscous as well as
+the inertial regime. Furthermore, to collapse the bridge growth for different capillary velocities on a single master
+curve, we use tc = 288Ki
+K3v
+µ3
+ρσ2θ2 and hc = 72Ki
+K2v
+µ2
+ρσ with Ki = 0.106 and Kv = 2.21 as previously obtained from similarity
+solutions of the thin-sheet equation [32]. Since we are only interested in the initial phase of the coalescence to limit
+finite size effects, we stop our simulations when h0 has reached 2/3 of the height of the lenses.
+In the viscous regime our simulations yield a linear bridge growth h0 ∼ t, followed by a crossover region that
+provides a smooth transition towards the h0 ∼ t2/3 dependence of the inertial regime (see Fig. 2). All simulations
+show very good agreement with the analytical solution of the thin-sheet equations. Noticeably, the numerical constants
+Ki and Kv from [32] yield an excellent collapse of the data sets, confirming that the thin-sheet equation is a good
+approximation to describe the coalescence dynamics in the case of small contact angles.
+
+5
+10
+3
+10
+1
+101
+103
+105
+107
+t/tc
+10
+3
+10
+1
+101
+103
+105
+h0/hc
+/ =11.6
+/ =5.8
+/ =0.696
+/ =0.348
+/ =1.16
+/ =2.9
+/ =0.0348
+/ =0.0116
+/ =0.00116
+/ =0.000116
+h0/hc = t/tc
+h0/hc = (t/tc)2/3
+theory
+Figure 2. Power law relation for the bridge growth in 2d covering the viscous as well as inertial regime (solid line: interpolation
+according to Eq. (16), dashed line: viscous theory, dotted-dashed line: inertial theory).
+0
+250
+500
+750
+1000
+1250
+1500
+1750
+2000
+y [Δx]
+0
+200
+400
+600
+z [Δx]
+0.5
+1.0
+1.5
+10-5 [Δx/Δt] 
+0
+250
+500
+750
+1000
+1250
+1500
+1750
+2000
+y [ x]
+0
+200
+400
+600
+z [ x]
+10-3 [Δx/Δt]
+1.0
+3.0
+5.0
+Figure 3. Flow field of viscous (σ/µ = 0.000116, left) and inertial (σ/µ = 0.348, right) liquid lens coalescence, where the grey
+scale of the velocity vectors represents the magnitude of the velocity vectors.
+The velocity field in the viscous regime is inherently dipolar and approaches a plug flow inside the liquid lens phase
+over time – see Fig. 3 (left panel) for a representative velocity field obtained from the simulations. While in the
+vicinity of the bridge minimum the flow field of the inertial regime is still dipolar (Fig. 3, right panel), two additional
+dipolar flow structures arise approximately at the center of each of the two initial liquid lenses. Furthermore, at larger
+distances from the bridge center fluid inertia causes the appearance of circulations in the wake of the retracting tips
+of the liquid lenses.
+In analogy to the assumptions of the thin sheet equation, Fig. 4 shows the profile uy(y, t) of the y-component of the
+velocity, averaged over the droplet extension along the z axis. Close to the bridge center (|ξ| < 1) the velocity profile
+is in good agreement with the prediction of the thin-sheet equation for the viscous as well as the inertial case. At
+larger distances to h0 (|ξ| > 1), however, the simulated velocity profile starts deviating from the thin-sheet solution.
+This effect can be attributed to the finite size of the lens as well as the difference in the treatment of the outer fluids:
+In contrast to the thin-sheet equation, our simulations include the full dynamics of the surrounding fluids with a finite
+viscosity. Thus, viscous damping in the surrounding fluids influences the velocity field inside the droplets.
+Next, we extend our simulations to the fully three-dimensional case (Fig. 5), where we use a system size of 768 ×
+2096 × 768 lattice nodes in x, y and z direction with periodic boundary conditions. The update of 1.2 · 109 lattice
+sites requires a considerable amount of computational resources. Therefore, the simulations were conducted on the
+JURECA Booster machine with 32, 768 Intel KNL cores using up to 3.4 million core-hours to generate a single data
+set.
+In analogy to the pseudo two-dimensional case, we initialize two equilibrated lenses with a contact angle of θ = 30◦
+(see Fig. 1) and adequate spacing to the domain boundaries. The growth of the bridge width reported in the left
+panel of Fig. 6 scales as w0 ∼ t1/2, which agrees with experiments [20, 21, 29, 31], analytical [4, 35] and numerical
+studies [52, 53] for freely suspended, respectively spherical droplets. The evolution of the bridge height h0, on the
+contrary, does not behave as in the quasi two-dimensional case (t2/3 scaling), but follows again the scaling h0 ∼ t1/2
+found for the width, as reported in the right panel of Fig. 6. This indicates that the thin-sheet equation for the
+2d case fails to describe the dynamics of the three-dimensional bridge growth. The scaling law is however not in
+
+6
+15
+10
+5
+0
+5
+10
+15
+= y
+h0
+0.75
+0.50
+0.25
+0.00
+0.25
+0.50
+0.75
+U = uy
+/kv
+t/tc=1.6 ⋅ 10-3
+t/tc=3.2 ⋅ 10-3
+t/tc=4.8 ⋅ 10-3
+t/tc=6.4 ⋅ 10-3
+theory
+10
+5
+0
+5
+10
+= y
+h0
+1.0
+0.5
+0.0
+0.5
+1.0
+U = uy 3 t1/3/(2ki)
+t/tc=6.5 ⋅ 102 
+t/tc=1.3 ⋅ 103
+t/tc=2.0 ⋅ 103
+t/tc=2.6 ⋅ 103
+theory
+Figure 4. Average profile of the y component of the velocity at different times in the viscous (left) and inertial (right) regimes
+compared to thin-sheet theory.
+1500
+1400
+300
+350
+x [Δx]
+400
+450
+600
+1300
+1200
+1100
+y [Δx]
+1000
+400
+z [Δx]
+900
+800
+700
+200
+600
+Figure 5.
+Snapshot of two coalescing liquid lenses in 3d.
+The snapshot is taken at t/tc = 285.8 (12,000 ∆t), where the
+connecting bridge has already developed for a capillary velocity σ/µ = 2.9 (inertial regime).
+contradiction to the experimental data shown in Ref. [32], where reasonably the transition region between the viscous
+and the inertial regime was observed. In the three-dimensional case the naive assumption of a decoupled width and
+height growth is clearly not satisfied. Since the two directions are strongly coupled, it is reasonable to expect that
+w0, which entails a larger amount of fluid than h0, is dominating the dynamics of the inertial regime for the whole
+bridge.
+In this case, we could not use hc as predicted by the analytical solution of the thin-sheet equation, and we settled
+for finding the best fitting value of hc for each data set. To check that the solution is not arbitrary, we plot the values
+of hc as a function of the ratio σ/µ of each data set, as reported in Fig. 7. The dependence is clearly of the type
+hc ∼ µ/σ. However, since hc can be expressed dimensionally in terms of surface tension and viscosity as hc ∼ µ2/(σρ),
+it is clear that this relation incorporates a (constant) prefactor with the dimensions of a kinematic viscosity.
+IV.
+CONCLUSION
+Liquid lens coalescence is an intrinsically multiscale problem and studying its scaling laws involves investigating
+surface tensions and viscosities that cover several orders of magnitude. Our simulation method - the color-gradient
+lattice Boltzmann method - has proven to deliver hydrodynamically consistent results for the required wide parameter
+ranges. This allows us to investigate the coalescence dynamics from the viscous to the inertial regime. For the pseudo
+two-dimensional case we find good agreement with the similarity solutions of the thin-sheet equation. In the viscous
+regime the bridge grows linearly with time and in the inertial regime, the bridge growth is proportional to t2/3.
+
+7
+10
+3
+10
+1
+101
+103
+105
+107
+t/tc
+10
+3
+10
+1
+101
+103
+105
+w0/wc
+/ =2.9
+/ =1.16
+/ =0.696
+/ =0.348
+/ =0.0348
+/ =0.0116
+/ =0.00116
+w0/wc = t/tc
+w0/wc = (t/tc)1/2
+w0/wc = (
+1
+t/tc +
+1
+(t/tc)1/2 )
+1
+10
+3
+10
+1
+101
+103
+105
+107
+t/tc
+10
+3
+10
+1
+101
+103
+105
+h0/hc
+/ =2.9
+/ =1.16
+/ =0.696
+/ =0.348
+/ =0.0348
+/ =0.0116
+/ =0.00116
+h0/hc = t/tc
+h0/hc = (t/tc)1/2
+h0/hc = (
+1
+t/tc +
+1
+(t/tc)1/2 )
+1
+Figure 6. Power law relation for the bridge growth in 3d covering the viscous as well as inertial limit (solid line: interpolation
+according to Eq. (16), dashed line: t, dotted-dashed line: t1/2). Left panel: bridge width w0(t); right panel: bridge height
+h0(t).
+10
+3
+10
+2
+10
+1
+100
+/
+10
+1
+100
+101
+102
+hc
+hfit
+c ( ,
+)
+hc
+( / )
+1.00
+Figure 7. Dependence of the best-fit hc on the capillary velocity in 3d. The dashed line represents the linear relation obtained
+by fitting the exponent of capillary velocity (σ/µ) to the data points.
+The three-dimensional coalescence simulations, on the contrary, deviate from the similarity solution of the thin-
+sheet equation exhibiting a t1/2 dependence. This can be explained by a strong coupling between the two directions
+and the involvement of a larger mass of fluid in the bridge width as compared to the bridge height. This makes the
+dynamics of the bridge width the dominant process.
+These results underline the necessity of a more generic theoretical framework for a more accurate understanding of
+the general coalescence process. In future studies, the influence of asymmetric properties of the liquid lenses on the
+coalescence dynamics could be investigated, for instance by extending the simulations to top-down asymmetric lenses
+or lenses with different viscosities or even non-Newtonian properties.
+ACKNOWLEDGMENTS
+We acknowledge Jacco Snoeijer and Michiel Hack for fruitful discussions. This work has received financial sup-
+port from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), within the priority program
+SPP2171 “Dynamic Wetting of Flexible, Adaptive, and Switchable Substrates”, projects HA-4382/11-1 and SE-
+3019/1-1 as well as SFB 1452 “Catalysis at liquid interfaces”, Project-ID 431791331.
+We also thank the J¨ulich
+
+8
+Supercomputing Centre for providing the necessary computing time.
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+

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+This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics
+1
+The effective diffusivity of ordered and freely
+evolving bubbly suspensions
+Aurore Loisy†‡, Aurore Naso and Peter D. M. Spelt
+Laboratoire de M´ecanique des Fluides et d’Acoustique,
+CNRS, Universit´e Claude Bernard Lyon 1, ´Ecole Centrale de Lyon, INSA de Lyon
+36 avenue Guy de Collongue, 69134 ´Ecully cedex, France
+(Received xx; revised xx; accepted xx)
+We investigate the dispersion of a passive scalar such as the concentration of a chemical
+species, or temperature, in homogeneous bubbly suspensions, by determining an effective
+diffusivity tensor. Defining the longitudinal and transverse components of this tensor with
+respect to the direction of averaged bubble rise velocity in a zero mixture velocity frame
+of reference, we focus on the convective contribution thereof, this being expected to be
+dominant in commonly encountered bubbly flows. We first extend the theory of Koch
+et al. (1989) (which is for dispersion in fixed beds of solid particles under Stokes flow) to
+account for weak inertial effects in the case of ordered suspensions. In the limits of low
+and of high P´eclet number, including inertial effect of the flow does not affect the scaling
+of the effective diffusivity with respect to the P´eclet number. These results are confirmed
+by direct numerical simulations performed in different flow regimes, for spherical or very
+deformed bubbles and from vanishingly small to moderate values of the Reynolds number.
+Scalar transport in arrays of freely rising bubbles is considered by us subsequently, using
+numerical simulations. In this case, the dispersion is found to be convectively enhanced at
+low P´eclet number, like in ordered arrays. At high P´eclet number, the Taylor dispersion
+scaling obtained for ordered configurations is replaced by the one characterizing a purely
+mechanical dispersion, like in random media, even if the level of disorder is very low.
+1. Introduction
+Bubble columns are commonly used in a broad range of technologies, notably in the
+chemical and biochemical industry. Simple bubble columns do not require active stirring
+and can therefore operate without interior moving parts. The large surface area between
+gas and liquid is useful for mass and species transfer, possibly involving chemical reactions
+(e.g., Deckwer 1992) such as in air-lift bioreactors, an example of which is in the treatment
+of wastewater. Bubble columns are also used for this reason in direct contact heat transfer
+(e.g., Hewitt et al. 1994). Besides offering a large surface area, rising bubbles agitate the
+liquid flow, which results in enhanced mixing that usually is desired, but this also poses
+a modelling difficulty. Similar mixing arises also in the diffusion through porous media in
+the presence of flow, which has been well studied previously, but mostly for fixed beds of
+particulates, often under creeping flow (e.g., Batchelor 1974; Koch & Brady 1985). Mixing
+in bubble columns is complicated further by the fact that liquid velocity fluctuations are
+coupled with the dynamics of (deformable) bubbles, usually beyond creeping flow (e.g.,
+Alm´eras et al. 2015).
+In the present study, we consider transport of a scalar (such as the concentration of
+† Present address: School of Mathematics, University of Bristol, University Walk, Bristol BS8
+1TW, United Kingdom.
+‡ Email address for correspondence: [email protected]
+arXiv:2301.00028v1  [physics.flu-dyn]  30 Dec 2022
+
+2
+A. Loisy, A. Naso, P. D. M. Spelt
+a chemical species, or the temperature) through incompressible bubbly flows. Gradients
+of temperature and concentration may, in general, induce fluid motion and influence the
+velocity field through changes in density and viscosity, or through the interface rheology.
+If these effects are small, as assumed herein, temperature and solute concentration can
+be considered as passive scalars. Although the arbitrary choice was made in this study
+to use the terminology of the mass transfer problem, the results carry over to thermal
+applications (upon assuming that effects of viscous heating can be ignored).
+Our present main interest is the formulation and closure of conservation equations and
+constitutive relations governing the dispersion of such a scalar in a bubbly suspension over
+scales (termed hereinafter the “macroscale”) that are much larger than the bubble size
+(termed hereinafter the “microscale”). Under the assumption of macroscale homogeneity
+and stationarity, scalar dispersion in multiphase systems can be described by a macroscale
+version of Fick’s (or Fourier’s) law which relates the macroscale scalar flux to the
+macroscale scalar gradient through an effective diffusivity tensor (or effective conductivity
+tensor in thermal applications) (Batchelor 1974; Koch & Brady 1985, 1987). This effective
+diffusivity is defined from an Eulerian perspective. Experimentally, scalar dispersion is
+usually investigated from a Lagrangian point of view. In the Lagrangian framework,
+the effective diffusivity is defined as the long-time limit of the time rate of change of a
+fluid tracer’s mean-square displacement, that is, as a measure of spread about the mean
+position. Koch & Brady (1987) demonstrated that the Lagrangian effective diffusivity
+is equivalent to the symmetric part of the Eulerian effective diffusivity, and that the
+antisymmetric part of the Eulerian effective diffusivity is associated with anisotropic
+microstructures.
+Scalar dispersion in a suspension of particulates (bubbles, drops, or rigid particles)
+results from two processes of very different nature: the diffusion by Brownian motion
+of the molecules, and the convection by the fluid velocity disturbances induced by the
+particulate motion. The relative importance of these two processes is measured by the
+P´eclet number Pe = Udb/D, where U is the characteristic velocity of the particulates
+relative to that of the system (defined in section 2), db is the characteristic size of the
+particulates, and D is the diffusivity of the bulk. In the limit Pe = 0, the effective
+diffusivity is purely diffusive and depends only on the particulate-to-bulk diffusivity ratio,
+possible discontinuity of the scalar at the interface, particulate volume fraction, and
+suspension microstructure (i.e., the positions, shapes, and orientations of the inclusions).
+This particular situation is essentially relevant to heat and electricity conduction in
+composite materials. When Pe ≫ 1, the dominant contribution to the effective diffusivity
+is due to convective mixing. This last regime is that generally encountered in bubbly flows.
+Recently Alm´eras et al. (2015) investigated experimentally the dispersion of a low-
+diffusive dye within a homogeneous swarm of high-Reynolds-number rising bubbles at
+Pe = O(106); herein we define the Reynolds number as Re = Udb/νc where νc is the
+kinematic viscosity of the liquid. They showed that scalar mixing primarily results from
+pseudo-turbulence, i.e., from the liquid agitation produced by bubble wake interactions,
+and can be modeled in a manner analogous to dispersion in shear-induced turbulence
+(Taylor 1921). Apart from the work of Alm´eras et al. (2015), the only other experimental
+investigation of mixing in homogeneous bubbly flows reported in the literature is the
+preliminary study of Mareuge & Lance (1995) which consists in a single data point.
+To the best of our knowledge, neither theoretical nor numerical investigations of scalar
+mixing in homogeneous bubbly flows have been reported thus far. Theoretical work is,
+however, available for other types of multiphase systems, and we shall review these now.
+The determination of such an effective diffusivity, at the macroscale, necessitates
+consideration of the conditions at the microscale. One class of analytical work is devoted
+
+The effective diffusivity of ordered and freely evolving bubbly suspensions
+3
+to the study of dilute systems with fixed random microstructure, for instance, as a model
+of a porous medium. In the absence of convection (Pe = 0), the analytical expression of
+the effective diffusivity is available in the dilute limit from analysis of the corresponding
+problem in conduction of heat or electricity through a dispersed medium (e.g., Maxwell
+(1873), Jeffrey (1973)). The problem of scalar dispersion in the presence of a bulk
+convective motion (Pe > 0) has been analyzed by Koch & Brady (1985) for Stokes
+flow through a random bed of fixed solid spheres. Using the method of conditional
+averaging pursued earlier by Hinch (1977), they carried out an asymptotic analysis
+in low volume fraction of the effective diffusivity for all values of the P´eclet number.
+Three mechanisms causing dispersion at high P´eclet number were identified: mechanical
+dispersion resulting from the stochastic velocity field in the bulk, which is independent
+of Brownian diffusion and grows as Udb, holdup dispersion in stagnant and recirculating
+regions which is proportional to U 2d2
+b/D, and boundary-layer dispersion which grows as
+Udb ln(Udb/D) near the solid particle surfaces.
+Another class of analytical studies assumes a periodic microstructure. For the pure dif-
+fusion problem (Pe = 0), analytical solutions have been derived for a composite material
+consisting of regularly arranged spheres embedded in a homogeneous matrix (Rayleigh
+1892; Sangani & Acrivos 1983), and the effect of anisotropy has been investigated by
+considering periodic arrangements of spheroidal inclusions (Kushch 1997; Harfield 1999).
+In the presence of convection (Pe > 0), the general theory of dispersion developed
+by Brenner (1980) and Brenner & Adler (1982) provides a consistent framework for
+determining the effective diffusivity in spatially periodic media. Koch et al. (1989) carried
+out explicit calculations for a periodic porous medium consisting of fixed solid particles
+arranged in a cubic lattice and embedded in a continuous phase under Stokes flow
+conditions. They showed that in ordered systems, the mechanical dispersion encountered
+in random media is absent, and that at high P´eclet number, either Taylor dispersion,
+growing as U 2d2
+b/D, or enhanced diffusion, which is proportional to D, is obtained
+depending on the direction of the mean flow relative to the lattice structure.
+In bubbly flows, the spatial arrangement of the inclusions evolves in time, the mi-
+crostructure of the suspension is unknown a priori, and Stokes flow is usually not
+applicable. For these reasons, prior analyses are, a priori, not applicable to bubbly
+suspensions. Nevertheless, we showed in prior work (Loisy et al. 2017) that the dy-
+namics of freely evolving bubbly suspensions at moderate Reynolds number shares some
+common features with that of ordered arrays of bubbles. It is therefore of fundamental
+interest to investigate, contrast and compare the mixing properties of ordered and freely
+evolving bubbly suspensions in light of prior asymptotic analyses for ordered and random
+arrangements of rigid particles.
+In this paper we investigate scalar dispersion, by determining the effective diffusivity in
+ordered and freely evolving bubbly suspensions, specifically, the contribution of bubble-
+induced velocity disturbances thereof. The prior work outlined above has established
+that in the systems studied therein, the effective diffusivity can be much larger than that
+in each of the fluids involved, even if the diffusivity in the two media is the same and
+the scalar is continuous at the surface of particulates. In view of the already significant
+number of parameters involved, we shall therefore adopt this restriction here. Such a
+simplified approach will not provide an accurate description of real bubbly flows, but
+should shed some light on the fundamental mechanisms of mixing in these systems.
+The paper is organised as follows. The theoretical framework and problem statement
+are provided in section 2. Our numerical approach to compute the effective diffusivity
+is presented, and followed by a description of the regimes and the range of parameter
+values that are investigated herein, in section 3. The first objective (in section 4) is to
+
+4
+A. Loisy, A. Naso, P. D. M. Spelt
+elucidate the role played by liquid inertia in ordered suspensions, using direct numerical
+simulation and analysis. The second objective (in section 5) is to investigate the effective
+diffusivity of freely evolving suspensions for a wide range of P´eclet numbers, to compare
+it with that obtained for ordered suspensions, and to evaluate the effect of introducing
+additional degrees of freedom in the system. Finally, the main results and perspectives
+of this work are provided in section 6.
+2. Problem statement
+The local evolution of the passive scalar c in each fluid is governed by
+∂c
+∂t + ∇ · q = 0
+(2.1a)
+where q is the flux of scalar given by
+q = uc − D∇c
+(2.1b)
+with u the fluid velocity and D the constant scalar diffusivity. We assume that the scalar
+and its gradient are continuous across the interface, and phase change is not considered
+in this study. Under these assumptions, no distinction between the phases is needed for
+the scalar transport, which is described by (2.1) in the entire system. We return to these
+restrictions in section 2.1 and in the Conclusions section; the objective here is to study
+this key basic reference problem.
+In the context of heat transfer, (2.1) derives from the energy balance upon neglecting
+viscous heating, in this case c would represent the temperature, continuous at the
+interface, and D the thermal diffusivity as defined by Fourier’s law, assumed to be equal
+in both gas and liquid. In the context of mass transfer, (2.1) describes the transport of
+a chemical species present at very low concentration c so that Fick’s law describes the
+conservation of mass, neglecting any difference in molecular diffusivity D and solubility
+of the species in the two phases. While the assumption of equal molecular diffusivities is
+never satisfied in real systems, the assumption of a unit dimensionless Henry’s constant
+is reasonably applicable to, e.g., carbon dioxide dispersion in the air-water system.
+The fluid motion in the gas and liquid is governed by the incompressible Navier-Stokes
+equations, which are coupled at the interface by the appropriate jump conditions, namely
+the continuity of velocity and of tangential traction across the interface, and a jump in
+normal traction due to surface tension.
+2.1. Macroscale description
+The problem we are concerned with here is the modeling of scalar transport at a
+macroscale, that is, at the scale whereat the suspension may be seen as a homogeneous
+continuum, without distinction between the two phases. In order to obtain such a
+macroscopic description, we consider an ensemble of realizations of the suspension,
+these realizations having the same macroscopic conditions (e.g., fluid properties, gas vol-
+ume fraction) but different microscopic configurations (e.g., bubble individual positions,
+shapes and velocities), and average over those realizations. In concrete terms, ensemble
+averaging would be realized by averaging over a large number of experiments run under
+identical macroscopic conditions. The ensemble-averaged transport equation is obtained
+from ensemble averaging the local transport equation (2.1). It reads
+∂⟨c⟩
+∂t
++ ∇ · ⟨q⟩ = 0,
+(2.2)
+
+The effective diffusivity of ordered and freely evolving bubbly suspensions
+5
+where ⟨ ⟩ denotes the ensemble average operator, and where the ensemble-averaged flux
+is given by
+⟨q⟩ = ⟨u⟩⟨c⟩ − D∇⟨c⟩ + ⟨u′c′⟩
+(2.3)
+where the velocity fluctuations are defined by u′ = u − ⟨u⟩ and the scalar fluctuations
+by c′ = c − ⟨c⟩. Under the restrictions set out above, the average flux consists of three
+contributions: (i) ⟨u⟩⟨c⟩ is the advection of the average scalar field at the average system
+velocity; (ii) −D∇⟨c⟩ is the diffusion of the average scalar field directly by the average
+scalar gradient; (iii) ⟨u′c′⟩ corresponds to the advection of the scalar fluctuations by the
+velocity fluctuations induced by bubble motion.
+When the suspension is statistically homogeneous and in a statistically stationary
+state, the linearity in c of the local flux (2.1b) results, in the presence of an imposed
+constant average scalar gradient, in a macroscale constitutive relation of the form (Koch
+& Brady 1985, 1987):
+⟨q⟩ = ⟨u⟩⟨c⟩ − Deff · ∇⟨c⟩
+(2.4)
+where Deff is a constant effective diffusivity tensor. Comparison of the effective diffusivity
+definition (2.4) with the average flux expression (2.3) yields the expression of the effective
+diffusivity. In order to reflect the contributions to the scalar flux identified above, it is
+customary to write the effective diffusivity as
+Deff = DI + Dconv
+(2.5)
+where
+Dconv · ∇⟨c⟩ = −⟨u′c′⟩
+(2.6)
+is the convective contribution arising from bubble-induced velocity fluctuations. For
+this model to be complete, one must find a closure relation for Dconv only in terms of
+macroscopic quantities appearing in the problem statement. We recall here that further
+contributions to the average flux (2.3) and hence to the effective diffusivity (2.5) arise
+if the diffusivity in the fluids are not the same, or if the concentration is discontinuous
+at fluid/fluid interfaces (e.g., Batchelor & O’Brien (1977); Koch & Brady (1985)). We
+return to the significance of this in section 6 below.
+2.2. Effective transport properties
+To determine the effective diffusivity for (unbounded) homogeneous bubbly suspen-
+sions, we represent such flows by the periodic repetition of a cubic unit cell containing a
+finite number Nb of freely moving bubbles of equal volume, building on our prior work on
+the dynamics of bubbles for this model system (Loisy et al. 2017). In the limit Nb = 1,
+one obtains a simple cubic array of bubbles, which is of interest as a model of perfectly
+ordered suspensions. The opposite limit of large Nb is of interest as a model of real
+suspensions, although convergence with the number of bubbles would have to be verified.
+We shall refer hereinafter to this setup with one bubble in the cell as an ordered array,
+and to that with more than one bubble in the unit cell as a free array.
+The bubbles rise under the sole effect of buoyancy. Herein, an upward-pointing primary
+axis e3 of the periodic arrangement is taken to be aligned with gravity (with the exception
+of the more general analysis presented in section 4.1). From symmetry arguments, and
+adopting a Cartesian coordinate system,
+Dconv =
+�
+�
+Dconv
+⊥
+Dconv
+12
+Dconv
+13
+Dconv
+12
+Dconv
+⊥
+Dconv
+13
+Dconv
+31
+Dconv
+31
+Dconv
+∥
+�
+�
+(2.7)
+
+6
+A. Loisy, A. Naso, P. D. M. Spelt
+where we have introduced the longitudinal and transverse components of the convective
+contribution to the effective diffusivity, denoted Dconv
+∥
+and Dconv
+⊥
+, respectively, and defined
+by
+Dconv
+∥
+= Dconv
+33
+and
+Dconv
+⊥
+= Dconv
+11
+= Dconv
+22 .
+(2.8)
+Our first goal is to characterize the effects of liquid inertia (through Re) on the
+dependence of Dconv on Pe for ordered suspensions (Nb = 1), thereby extending prior
+work on dilute ordered arrays of rigid spheres in Stokes flow conditions (Koch et al.
+1989). Our second goal is to evaluate the effect of introducing additional degrees of
+freedom in the system (through increasing Nb), and to investigate the dependence of
+Dconv on Pe in freely evolving suspensions (sufficiently large Nb). As we found the off-
+diagonal components to be zero in all configurations that we investigated, only results
+for the longitudinal and the transverse components of Dconv will be presented.
+In dimensionless groups, we shall use as characteristic length scale the bubble size db,
+which is defined, since bubbles are deformable, as the (equivalent) diameter of a sphere of
+the same volume. The characteristic velocity U is taken here as the bubble rise velocity
+in the frame of the suspension (the so-called drift velocity ⟨U⟩ = ⟨u⟩d − ⟨u⟩, where the
+first term is the volume average of velocity on the disperse phase only and the second
+one is the same average in the entire system). As already mentioned, a key dimensionless
+group appearing in the scalar transport problem is the P´eclet number Pe = Udb/D which
+compares advective and diffusive transport. Our main objective is to elucidate the effect
+of the value of Pe on the effective diffusivity using analytical and numerical methods.
+The effective diffusivity necessarily also depends on the gas volume fraction φ =
+(Nbπd3
+b)/(6h3) (h is the linear size of the unit cell); the analytical and computational
+methods used here pose some restrictions on the range of φ values that can be studied
+herein, we postpone discussion of that to the pertinent sections below. We also consider
+the effects of the number of bubbles in the periodic cell, Nb, which affects the order in
+the suspension: Nb = 1 corresponds to a cubic array, whereas more bubbles results in
+a different microstructure (the latter term encompasses all the information about the
+statistical distribution of the bubble positions, shapes, orientations, etc.). Since scalar
+transport is coupled to momentum transport, the bubble Reynolds number Re = Udb/νc
+may also play a significant role that will be investigated here as well. The ranges of φ,
+Nb and Re studied here are summarized in table 1.
+As the bubbly flows we consider are buoyancy-driven, a difficulty arises from the fact
+that U is a priori unknown, and depends in a complex manner on Nb, φ, the density
+and viscosity ratios between both phases, the Archimedes (or Galileo) number Ar =
+�
+ρc|ρd − ρc|gd3
+b/µc, and the Bond (or E¨otv¨os) number Bo = |ρd − ρc|gd2
+b/γ, where the
+subscripts d and c refer to the disperse (gas) and continuous (liquid) phases, respectively,
+g is the magnitude of the gravitational acceleration, ρ denotes density, µ is the dynamic
+viscosity, and γ is the surface tension. In most bubbly flows of practical relevance, the
+gas-to-liquid density and viscosity ratios are vanishingly small. Their precise values are
+not important from a physical point of view as long as they are small enough; in the
+simulations, the gas-to-liquid density and viscosity ratios were set to ρd/ρc = 10−3 and
+µd/µc = 10−2, respectively. The dependence of U on (Ar, Bo, φ, Nb) has been addressed
+in Loisy et al. (2017) and is not further discussed here. In the present study, we shall
+therefore assume that U is known.
+
+The effective diffusivity of ordered and freely evolving bubbly suspensions
+7
+3. Methodology
+For convenience of numerical implementation, we reorganise the problem formulation by
+introducing the decomposition
+c = ¯c + ˜c
+(3.1)
+where ¯c is the imposed constant linear scalar field
+¯c = ∇⟨c⟩ · x.
+(3.2)
+The advantage of this decomposition is that the disturbance field ˜c is then spatially
+periodic. The governing equation for this disturbance field is
+∂˜c
+∂t + ∇ · (u˜c) − ∇ · (D∇˜c) = −u · ∇⟨c⟩
+(3.3)
+which is the equation we integrate numerically. The convective contribution to the
+effective diffusivity is then calculated from
+Dconv · ∇⟨c⟩ = −⟨u′˜c⟩
+(3.4)
+which can be shown to be equivalent to (2.6). In this expression, ⟨ ⟩ is defined as
+an ensemble average operator, as above. For statistically homogeneous and stationary
+systems, as considered here, it is inferred from ergodicity that ensemble averaging is
+identical to volume and time averaging. As a consequence, Dconv is computed from (3.4)
+with the ensemble average being replaced in practice by a volume average combined with
+a time average over an appropriate time period.
+3.1. Numerical method
+Thus, the components of Dconv are obtained from direct numerical simulations (DNS)
+by imposing a constant linear scalar field ¯c and determining the resulting periodic
+disturbance scalar field. Two distinct simulations are required to fully determine the
+five independent components of Dconv: in one simulation, ∇¯c = e3, which yields Dconv
+13
+and Dconv
+∥
+, in the other simulation, ∇¯c = e1, which yields Dconv
+⊥
+, Dconv
+12 , and Dconv
+31 . The
+off-diagonal components of Dconv were found to be zero (up to computer accuracy for
+ordered arrays, and statistical uncertainty for free arrays) for all the sets of parameters
+we considered, and therefore will not be shown.
+The numerical methods employed to solve the two-phase flow have been described in
+detail in Loisy et al. (2017). In short, we employ a standard projection method (Chorin
+1968) to integrate the incompressible Navier-Stokes equations, a level-set method (e.g.,
+(Sussman et al. 1994)) to capture the moving gas-liquid interface, and surface tension is
+accounted for using the continuum surface force model (Brackbill et al. 1992).
+Our algorithm proceeds iteratively through the following steps:
+(i) The position of the interface is first advanced in time according to the modi-
+fied level-set method of Sabelnikov et al. (2014) using a third-order total-variation-
+diminishing (TVD) Runge-Kutta scheme. The level-set function is then reinitialized using
+the procedure of Russo & Smereka (2000), and a correction is finally applied to enforce
+volume conservation.
+(ii) The scalar transport equation (3.3) is advanced by using a mixed Crank-
+Nicolson/third-order Adams-Bashforth time-stepping scheme.
+(iii) The time integration of the incompressible Navier-Stokes equations is then carried
+out using a mixed Crank-Nicolson/third-order Adams-Bashforth scheme and consists
+in the combination of a predictor step, where a temporary velocity field is estimated by
+
+8
+A. Loisy, A. Naso, P. D. M. Spelt
+case Bo
+Ar
+Nb
+φ
+Re
+bubble shape
+S0
+0.38 0.15 1
+0.002 0.00164 spherical
+S1
+0.38 5.03 1
+0.002 1.72
+spherical
+C
+243
+15.2 1
+0.002 9.44
+skirted
+E1
+2.0
+29.9 1
+0.002 39.9
+ellipsoidal
+E1
+2.0
+29.9 [1, 12] 0.024 ≈ 30
+ellipsoidal
+Table 1. Simulated flow configurations: Bo and Ar define the flow regime, Nb is the number
+of free bubbles in the unit cell, φ is the gas volume fraction. The resulting bubble Reynolds
+number (Re) and shape are also provided.
+ignoring the effect of pressure, and of a corrector step, where the velocity field is corrected
+by the pressure gradient term computed from the divergence-free condition.
+Spatial discretization relies on a mixed finite difference/finite volume approach on a
+fixed, staggered, Cartesian grid. Second-order centered schemes are generally employed,
+except for advective terms which are discretized using fifth-order weighted-essentially-
+nonoscillatory (WENO) schemes.
+Results of numerical tests are presented in the Appendix.
+3.2. Parametric study
+Four different flow regimes, as defined by the set (Ar, Bo), are considered here. These
+are described in table 1, and have been studied in Loisy et al. (2017) (the same case
+code names are used). In case S0, the bubbles are spherical and the Reynolds number is
+vanishingly small, which approaches Stokes flow conditions. In case S1, the bubbles are
+(nearly) spherical and Re ≳ 1. In case C, the bubbles are skirted, and Re ≈ 8. In case
+E1, the bubbles are ellipsoidal, and Re ≈ 30 − 40.
+Ordered arrays of bubbles in these four flow regimes have been considered for the
+smallest volume fraction numerically accessible (value provided in table 1). After a
+transient regime, all ordered suspensions considered here are in a strictly steady state (for
+the flow and the scalar) during which the results presented in section 4 were obtained.
+Simulations of scalar transport in free arrays have been performed for 2 ⩽ Nb ⩽ 12 in
+case E1 at φ = 2.4 %. In these conditions, coalescence is indeed absent (it does occur at
+larger φ), whereas simulations at lower φ for free arrays are excessively expensive for the
+method and facilities used. In this regime, the system is in an unsteady but statistically
+stationary state (for the flow and the scalar), during which the statistics presented in
+section 5 have been measured. For each of these configurations (Ar, Bo, φ, Nb), the
+drift velocity (and thereby the Reynolds number) is known from Loisy et al. (2017). This
+allowed us to impose the P´eclet number a priori.
+The numerical simulation results for ordered arrays are compared with the results of
+analysis at small (but possibly finite) Reynolds number and small volume fraction.
+4. Ordered suspensions
+We examine in this section the dispersion of a passive scalar in ordered suspensions of
+deformable bubbles. Our main objective here is to elucidate the effects of inertia on
+dispersion, using theoretical analysis and numerical simulation.
+
+The effective diffusivity of ordered and freely evolving bubbly suspensions
+9
+4.1. Asymptotic analysis
+We first determine analytically the convective contribution to the effective diffusivity
+of ordered suspensions of spherical fluid particulates (bubbles or drops). The Reynolds
+number of the particulates is assumed to be small so that the Navier-Stokes equations
+can be approximated by the Oseen equations.
+4.1.1. General solution
+An ordered array of particulates translating at a drift velocity U is equivalent to an
+ordered array of fixed particulates immersed in a viscous fluid moving with an average
+system velocity ⟨u⟩ = −U. The centers of the particulates are located on the nodes of a
+simple cubic lattice:
+rn = h (n1e1 + n2e2 + n3e3)
+n1, n2, n3 = 0, ±1, ±2, . . .
+(4.1)
+where h is the lattice spacing and ei are the unit vectors aligned with the primitive axes
+of the cubic lattice. In the dilute limit (db/h ≪ 1), the action of these particulates on the
+fluid can be represented by point forces −f. The convective contribution to the effective
+diffusivity arising from the far field has been derived by Koch et al. (1989) for an ordered
+array of rigid spheres in the Stokes flow regime. In what follows we extend their result
+to the case of spherical fluid particulates at small but finite Re.
+When Pe ≪ 1, the convective contribution to the effective diffusivity arising from the
+far field can be approximated by (Koch et al. 1989):
+Dconv
+D
+=
+�
+k̸=0
+k2ˆu′(k)ˆu′(−k)
+(2π)2k4D2 + (U · k)2 ,
+(4.2)
+where the summation is over all vectors k in the reciprocal lattice
+k = 1
+h (n1e1 + n2e2 + n3e3)
+(4.3)
+and where ˆu′ is the three-dimensional Fourier transform of the velocity disturbance
+u′ = u − ⟨u⟩. In Oseen flow past an ordered array of point particulates, ˆu′ is given by
+ˆu′(k) =
+f · (kk/k2 − I)
+(2πk)2h3µc + i2πh3ρcU · k
+k ̸= 0,
+(4.4)
+where f is the hydrodynamic force exerted by the ambient fluid on a particulate. In the
+dilute limit, f can be approximated by the Oseen drag exerted on a single spherical fluid
+particulate:
+f = Ff 0,Stokes
+(4.5)
+where f 0,Stokes is the Stokes drag on that particulate (Hadamard 1911; Rybczynski 1911):
+f 0,Stokes = −2πµ∗µcdbU,
+with µ∗ = µc + 3µd/2
+µc + µd
+,
+(4.6)
+and where F accounts for the finite-Re correction to the Stokes drag (Brenner & Cox
+1963):
+F = 1 + 1
+8µ∗Re.
+(4.7)
+The convective contribution to the effective diffusivity of a dilute ordered array of fluid
+particulates in Oseen-flow conditions is therefore:
+Dconv
+D
+=
+µ∗2
+(2π)2
+d2
+b
+h2 F 2C,
+(4.8a)
+
+10
+A. Loisy, A. Naso, P. D. M. Spelt
+regime
+∥Dconv∥/(DF 2d2
+b/h2)
+Peh = Uh/D Reh = ρcUh/µc
+if ∃rn | U ⊥ rn if ∄rn | U ⊥ rn
+Peh ≪ 1
+Reh ≪ 1
+Pe2
+h
+Pe2
+h
+Reh ≫ 1
+Pe2
+h
+Pe2
+h/Re2
+h
+Peh ≫ 1
+Reh ≪ 1
+Pe2
+h
+1
+Reh ≫ 1
+Pe2
+h
+1/Re2
+h
+Table 2. Asymptotic order of ∥Dconv∥ depending on Peh, Reh, and on the orientation of the
+mean flow relative to the real lattice, based on the solution (4.8), derived for an ordered array
+of point particulates in Oseen flow conditions (F is the Oseen drag divided by the Stokes drag).
+where C is the dimensionless tensor:
+C =
+�
+k∗̸=0
+�
+U ∗ ·
+�k∗k∗
+k∗2 − I
+��2
+k∗2
+�
+(2π)2k∗4
+Pe2
+h
++ (U ∗ · k∗)2
+��
+1 + Re2
+h(U ∗ · k∗)2
+(2π)2k∗4
+�
+(4.8b)
+with U ∗ = U/U, k∗ = kh, Reh = ρcUh/µc, and Peh = Uh/D. The solution given by
+Koch et al. (1989) (equation (4.5) therein) for rigid spheres and Stokes flow is recovered
+in the limit Re → 0 and µd/µc → ∞.
+The tensor C only depends on Peh, Reh, and on the orientation of U relative to the
+reciprocal lattice (which structure is, for cubic arrays, identical to that of the direct
+lattice). As highlighted by Koch et al. (1989), the asymptotic behavior of C, and hence
+of Dconv, depends on whether there exists any k such that U · k = 0, that is, on whether
+there exists any separation vector rn in the real space which is perpendicular to U.
+The asymptotic behavior of ∥Dconv∥, where ∥ ∥ denotes the tensorial Frobenius norm, is
+provided in table 2. The results show that the dependence of ∥Dconv∥ on Pe in the limits
+Peh ≪ 1 and Peh ≫ 1 is, qualitatively, not affected by (weak) inertial effects.
+4.1.2. Application to ordered arrays rising vertically
+Let us now come back to our original problem of an ordered array of particulates rising
+under the effect of buoyancy. The gravitational acceleration is oriented along a primary
+axis of the array, g = −ge3, and although this is not the only possible solution (see,
+e.g., Loisy et al. (2017)), we restrict the analysis to the simplest case of bubbles rising
+vertically. In this case the hydrodynamic force exerted by the fluid on a particulate is
+parallel to the drift velocity, and, since this force balances the buoyancy force at steady
+state, F is related to U through
+F = U0,Stokes
+U
+(4.9)
+where U0,Stokes is the terminal velocity of an isolated spherical fluid particulate in Stokes
+flow:
+U0,Stokes = 1
+12
+|ρc − ρd|gd2
+b
+µ∗µc
+,
+with µ∗ = µc + 3µd/2
+µc + µd
+.
+(4.10)
+Note that F can also be expressed in terms of commonly employed dimensionless groups:
+F =
+1
+12µ∗
+Ar 2
+Re .
+(4.11)
+
+The effective diffusivity of ordered and freely evolving bubbly suspensions
+11
+Peh
+D||
+conv (D F2 Pe2)  x103
+(a)
+10−1
+100
+101
+102
+103
+104
+105
+3
+3.1
+3.2
+3.3
+3.4
+Re = 10−8
+Re = 10−6
+Re = 10−4
+Re = 10−2
+Peh
+D⊥
+conv (D F2 db
+2 h2)
+(b)
+10−1
+100
+101
+102
+103
+104
+105
+10−12
+10−10
+10−8
+10−6
+10−4
+10−2
+100
+∝ Pe2
+Re = 10−8
+Re = 10−6
+Re = 10−4
+Re = 10−2
+Figure 1. Longitudinal (a) and transverse (b) components of Dconv as a function of the P´eclet
+number based on the lattice spacing (Peh = Uh/D) for ordered arrays of point particulates at
+various small but finite Reynolds numbers (U = Ue3, db/h = 10−6, and F is given by (4.9)).
+Note that in (a), Dconv
+∥
+is compensated by Pe2.
+In the “sedimentation” problem considered here, F is generally not known (as U is
+generally not known): it is a non-trivial function of the flow regime and volume fraction
+which reduces to (4.7) when φ → 0 and when Oseen-flow approximation is applicable.
+The longitudinal and transverse components of the convective contribution, Dconv
+∥
+and
+Dconv
+⊥
+respectively, have been calculated from (4.8) for db/h = 10−6 as a function of Peh
+for various Re < 1. This very low value of db/h is required to allow Peh ≫ 1 while
+satisfying the condition Pe = Peh db/h ≪ 1 under which the analytical solution has
+been derived. The results, shown in figure 1, indicate that the asymptotic dependences
+of Dconv
+∥
+and Dconv
+⊥
+on Pe are independent of Re. The sole effect of inertia is to modify
+the proportionality constants (by a substantial amount for the transverse component
+though).
+In the limit of low Peh (say, Peh < 101), both the transverse and the longitudinal com-
+ponents of Dconv exhibit a quadratic dependence on the P´eclet number (Dconv
+⊥,∥ ∝ DPe2).
+In this regime, diffusion is much faster than convection. As the scalar is advected by
+velocity disturbances, it rapidly spreads out owing to diffusion, and convective dispersion
+(measured through Dconv) is influenced by both mechanisms. This regime corresponds to
+the “convectively enhanced dispersion” regime in Koch et al. (1989).
+In the limit of high Peh (say, Peh > 103), the transverse component of Dconv is
+independent of the P´eclet number (Dconv
+⊥
+∝ D) whereas its longitudinal component
+grows quadratically with the P´eclet number (Dconv
+∥
+∝ DPe2). In this regime, convection
+dominates, but owing to the spatial periodicity of the flow, convective dispersion is
+obtained only if molecular diffusion across streamlines is considered (Koch et al. 1989).
+This regime is termed “Taylor dispersion” owing to the formal analogy, pointed out by
+Brenner (1980), with one-dimensional shear-induced Taylor dispersion in a capillary tube.
+We emphasize that the expression (4.8) has been derived from the approximation (4.2),
+the validity of which is established only for Pe ≪ 1 (which is, in practice, of limited use).
+Using symmetry arguments, Koch et al. (1989) (section 4.2 therein) showed that in the
+limit Pe ≫ 1, Taylor dispersion is obtained if the average flow is perpendicular to a set of
+planes of both translational and reflectional symmetry, such as Stokes flows parallel to the
+primary axis of an ordered array of spheres. Taylor dispersion is then easily understood
+by remarking that, owing to the symmetries of the flow, a fluid tracer particle entering the
+unit cell at one point, say x, exits the cell at the equivalent point in the next cell, that is,
+
+12
+A. Loisy, A. Naso, P. D. M. Spelt
+Peh
+D||
+conv (D F2 Pe2)  x103
+100
+101
+102
+103
+104
+2.8
+3
+3.2
+3.4
+3.6
+(a)
+Re = 0.0, spherical   (S0)
+Re = 1.7, spherical   (S1)
+Re = 9.4, skirted       (C)
+Re = 40 , ellipsoidal (E1)
+Peh
+D⊥
+conv (D F2 db
+2 h2)
+100
+101
+102
+103
+104
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+(b)
+∝ Pe2
+Figure 2. Longitudinal (a) and transverse (b) components of Dconv as a function of the P´eclet
+number based on the lattice spacing (Peh = Uh/D) for ordered arrays in various flow regimes
+at small volume fraction (φ = 0.2 %). The normalizations of Dconv
+∥,⊥ are those suggested by the
+asymptotic analysis (identical to those used in figure 1), and F is given by (4.9). The lines are
+drawn to guide the eyes. Note that in (a), Dconv
+∥
+is compensated by Pe2.
+x+he3, so that dispersion can only occur if diffusion across streamlines is present (Koch
+et al. 1989). In the presence of inertial effects, the reflectional symmetry is lost, hence
+this argument does not hold. Koch et al. (1989) also demonstrated that, for Stokes flow,
+the solution for Pe ≪ h/db is identical, at lowest order, to that obtained for Pe ≪ 1
+(section 4.3 therein, note that their Pe corresponds to Peh in our notations). Such a
+demonstration for Oseen flow will not be attempted here. Instead, the range Pe ⩾ 1 will
+be explored using direct numerical simulations.
+4.2. Numerical results
+The above analysis provides explicit expressions of Dconv
+∥
+and Dconv
+⊥
+. These are valid for
+spherical bubbles rising at Re < 1 (strictly speaking, at a Reynolds number sufficiently
+small to assume Oseen flow, in terms of Archimedes and Bond numbers this regime would
+be reached for Bo < 1 and Ar ≲ 1), and in the limits φ → 0 and Pe ≪ 1. We shall now
+determine using numerical simulations whether these restrictions can be relaxed, and if
+so, to which extent.
+We examine the case of suspensions at low (but not vanishing) volume fraction
+in order to approach the dilute limit assumption, and to focus on the sole effect of
+inertia. The longitudinal and transverse components of the convective contribution to
+the effective diffusivity have been computed for h/db = 6.4, which corresponds to a
+gas volume fraction of φ = 0.2 % (the smallest volume fraction accessible with the
+method and facilities used), for each of the four flow regimes listed in table 1, and Pe
+has been varied from 10−1 to 103. The results are shown in figure 2 as a function of
+Peh, the P´eclet number based on the lattice spacing, which is the parameter governing
+the transition between the two asymptotic limits (see section 4.1 and figure 1). The
+different colors, symbols and line styles depict the different flow regimes (the lines are
+drawn to guide the eyes). Qualitatively, figure 2 bears a striking resemblance to figure 1,
+even for case C (skirted bubbles): analysis and simulations yield similar dependences
+of Dconv
+∥,⊥ on Pe and qualitatively comparable effects of increasing Re. At low P´eclet
+number (Peh ≲ 101), dispersion occurs primarily by molecular diffusion and convective
+mixing grows quadratically with Pe in both the longitudinal and the transverse directions
+
+The effective diffusivity of ordered and freely evolving bubbly suspensions
+13
+Pe
+D||
+conv D||
+conv,anal
+10−1
+100
+101
+102
+103
+0.9
+0.95
+1
+1.05
+1.1
+1.15
+1.2
+(a)
+Re = 0.0, spherical   (S0)
+Re = 1.7, spherical   (S1)
+Re = 9.4, skirted       (C)
+Re = 40 , ellipsoidal (E1)
+Pe
+D⊥
+conv D⊥
+conv,anal
+10−1
+100
+101
+102
+103
+0
+2
+4
+6
+8
+(b)
+Figure 3. Numerical solution Dconv divided by the analytical solution Dconv,anal as a function of
+the P´eclet number based on the bubble diameter (Pe = Udb/D) for ordered arrays in various flow
+regimes at small volume fraction (φ = 0.2 %): longitudinal (a) and transverse (b) components.
+Dconv,anal is given by (4.8).
+(Dconv
+∥,⊥ ∝ DPe2). At high P´eclet number (Peh ≳ 103), Taylor dispersion is the dominant
+process, with very efficient mixing in the flow direction (Dconv
+∥
+∝ DPe2) and negligible
+mixing in the transverse one (Dconv
+⊥
+∝ D). Inertial effects and bubble deformation only
+affect the proportionality constants, rather weakly for Dconv
+∥
+but substantially for Dconv
+⊥
+.
+To allow a quantitative comparison between the DNS and the analysis, we present in
+figure 3 the ratio of Dconv
+∥,⊥ to Dconv,anal
+∥,⊥
+where Dconv,anal
+∥,⊥
+is given by (4.8) with F computed
+directly from its definition (4.9). As the range of validity of the analysis is defined in terms
+of Pe (Pe ≪ 1, with Pe the P´eclet number based on the bubble diameter), the data are
+presented here as a function of Pe rather than Peh. For the longitudinal component, the
+numerical solution does not deviate by more than 5 % from the theoretical prediction,
+as can be seen from figure 3(a). The fact that the low-Pe, Oseen-flow analysis yields
+accurate predictions for Dconv
+∥
+at Pe = 103 and Re = O(10) is not surprising, as the
+behavior of Dconv
+∥
+/(DF 2Pe2) is rather insensitive to both the flow regime and the P´eclet
+number (as shown in Fig. 1(a) and 2(a), this quantity does not vary more than 15% for
+the cases studied). We conclude that, at small volume fraction, Dconv
+∥
+can be predicted
+within ±5 % from (4.8) at any P´eclet number up to 103 and any Reynolds number up
+to 40, even when the bubbles are strongly deformed. For the transverse component, the
+asymptotic analysis underpredicts the value of Dconv
+⊥
+at high P´eclet number, even for
+Re ≲ 1. As a consequence, Dconv
+⊥
+cannot be accurately estimated from our analytical
+solution when the assumptions underlying its derivation are not satisfied. It must be
+kept in mind though that this component varies much more than the longitudinal one
+between the regimes of small and large P´eclet numbers, and is much more sensitive to
+the flow regime (Re, shape), which means that its value is more difficult to predict. In all,
+it is worth stressing that the asymptotic analysis yields the correct qualitative behavior
+and order of magnitude for Dconv
+⊥
+at least up to Pe = 103 and Re ≈ 10, even for strongly
+deformed bubbles. Finally, we emphasize that we found Dconv
+∥
+/Dconv
+⊥
+≳ 102, so the most
+important component of the effective diffusivity tensor is the longitudinal one, except in
+situations where there is no longitudinal component of the gradient of the scalar on the
+macroscale.
+To illustrate the dispersion regimes at low and high P´eclet number, we present in
+figure 4 and figure 5 visualizations of the scalar fluctuation field c′ used to compute
+
+14
+A. Loisy, A. Naso, P. D. M. Spelt
+Re = 0.0 (S0)
+Pe = 10−1
+Re = 1.7 (S1)
+Re = 9.4 (C)
+Re = 40  (E1)
+−0.04
+−0.03
+−0.02
+−0.01
+0.00
+c’
+Pe = 103
+−300
+−200
+−100
+0
+c’
+Figure 4. Scalar fluctuation field c′ associated with Dconv
+∥
+, shown in a vertical symmetry plane
+passing through the center of a bubble, for ordered arrays in various flow regimes at Pe = 10−1
+(left) and Pe = 103 (right). The imposed scalar field ¯c increases linearly within the cell from
+bottom to top (φ = 0.2 %, the entire cell is shown, and gravity is pointing downward).
+
+The effective diffusivity of ordered and freely evolving bubbly suspensions
+15
+Re = 0.0 (S0)
+Pe = 10−1
+Re = 1.7 (S1)
+Re = 9.4 (C)
+Re = 40  (E1)
+−0.002
+0.000
+0.002
+c’
+Pe = 103
+−0.4
+−0.2
+0.0
+0.2
+0.4
+c’
+Figure 5. Scalar fluctuation field c′ associated with Dconv
+⊥
+, shown in a vertical symmetry plane
+passing through the center of a bubble, for ordered arrays in various flow regimes at Pe = 10−1
+(left) and Pe = 103 (right). The imposed scalar field ¯c increases linearly within the cell from
+left to right (φ = 0.2 %, the entire cell is shown, and gravity is pointing downward).
+
+16
+A. Loisy, A. Naso, P. D. M. Spelt
+Pe
+D||
+conv D
+(a)
+10−1 100
+101
+102
+103
+104
+105
+106
+10−4
+10−2
+100
+102
+104
+106
+108
+1010
+∝ Pe2
+∝ Pe
+Nb=
+1
+2
+3
+5
+8
+12
+Pe
+D⊥
+conv D
+(b)
+10−1 100
+101
+102
+103
+104
+105
+106
+10−6
+10−4
+10−2
+100
+102
+104
+106
+108
+∝ Pe2
+∝ Pe
+Figure 6. Longitudinal (a) and transverse (b) components of Dconv as a function of the P´eclet
+number for various numbers of free bubbles Nb in the unit cell (Nb = 1 corresponds to an ordered
+array). Symbols other than purple stars: DNS (Re ≈ 30, φ = 2.4 %); purple stars: experimental
+data of Alm´eras et al. (2015) (Re ≈ 700, φ ≈ 2.4 %). A spatial resolution of db/∆x = 20 was
+used for Nb > 1, the effect of increasing resolution to db/∆x = 30 is illustrated by the filled red
+squares for Nb = 8 and Pe ≈ 103 (db is the bubble volume-equivalent diameter and ∆x is the
+grid spacing).
+Dconv
+∥
+and Dconv
+⊥
+, respectively. In each of these figures, the field of c′ is represented for
+each flow regime in a vertical symmetry plane passing through the center of a bubble
+for Pe = 10−1 (left) and Pe = 103 (right), and the Reynolds number increases from
+top to bottom. The field of c′ associated with Dconv
+∥
+, shown in figure 4, exhibits similar
+features at low and high Pe. In contrast, the field of c′ associated with Dconv
+⊥
+, represented
+in figure 5, is qualitatively different in these two limits. This illustrates qualitatively
+why the regimes at low and high Pe are similar for Dconv
+∥
+(Dconv
+∥
+∝ Pe2), whereas the
+scaling laws identified for Dconv
+⊥
+are different in both limits (see figure 2). In addition, the
+Reynolds number and the bubble shape affect the fore-and-aft symmetry and the details
+of c′, but not its essential features, which results in quantitative but not qualitative effects
+on Dconv
+∥
+and Dconv
+⊥
+.
+5. Freely evolving suspensions
+We examine in this section scalar mixing in freely evolving suspensions as represented
+by the periodic repetition of a unit cell containing several independent bubbles (“free
+arrays”). Our objective here is threefold: (i) to investigate the effective diffusivity of
+freely evolving suspensions at small and high P´eclet numbers, (ii) to compare and contrast
+these results with those obtained in ordered systems, and (iii) to evaluate the effect of
+the system size (number of bubbles in a unit cell, Nb).
+For that purpose, we considered a single flow regime (ellipsoidal bubbles at Re = O(10),
+corresponding to case E1 in table 1) at intermediate volume fraction (φ = 2.4 %) and
+explored the effect of varying the number of free bubbles Nb on the dependence of Dconv on
+the P´eclet number. Due to the multiplicity of simulations involved and to their duration
+(typically several months on 64 cores), only a few different values of Nb belonging to a
+rather limited range have been considered (namely Nb = {2, 3, 5, 8, 12} in the simulations
+for the determination of Dconv
+∥
+, and Nb = {2, 8} in those for Dconv
+⊥
+). For the same reason,
+investigations of the effects of volume fraction and flow regime could not be undertaken.
+The longitudinal and transverse components of Dconv are plotted in figure 6 as a
+
+The effective diffusivity of ordered and freely evolving bubbly suspensions
+17
+Figure 7. Instantaneous scalar fluctuation field c′ associated with Dconv
+∥
+for a free array of 8
+bubbles, at Pe = 10−1 (left) and Pe = 106 (right). The gradient of ¯c is vertical (the entire cell
+is shown, and gravity is pointing downward).
+Figure 8. Instantaneous scalar fluctuation field c′ associated with Dconv
+⊥
+for a free array of 8
+bubbles, at Pe = 10−1 (left) and Pe = 106 (right). The gradient of ¯c is horizontal (the entire
+cell is shown, and gravity is pointing downward).
+function of the P´eclet number for various values of Nb. Note that a very wide range of
+P´eclet numbers is considered. Convergence of Dconv
+∥
+with the system size is very fast: the
+values of Dconv
+∥
+are essentially independent of the number of free bubbles for 2 ⩽ Nb ⩽ 12
+at all P´eclet numbers. This suggests that Dconv
+∥
+is independent of the system size Nb,
+although this would need to be confirmed by considering larger values of this parameter.
+Our data for Dconv
+⊥
+suggest that convergence with Nb is slower for this quantity, especially
+at high P´eclet number, although conclusions can hardly be drawn on this point due to
+the few values of Nb considered.
+We first examine the dependence of Dconv on the P´eclet number in free arrays of
+bubbles (Nb > 1). At small Pe, Dconv
+∥,⊥ ∝ DPe2, whereas at high Pe, Dconv
+∥,⊥ ∝ DPe = Udb.
+Note that the scaling at high Pe is expected from a simple dimensional analysis in a
+convection-dominated regime where diffusion plays no role. This regime corresponds to
+the “mechanical dispersion” regime in Koch & Brady (1985). The different dispersion
+regimes at low and high Pe can also be identified from the features of the scalar
+fluctuation field c′. Instantaneous snapshots of c′ associated with Dconv
+∥
+and Dconv
+⊥
+are
+
+18
+A. Loisy, A. Naso, P. D. M. Spelt
+shown in figure 7 and figure 8, respectively, for an array of 8 free bubbles at Pe = 10−1
+(left) and at Pe = 106 (right). For a given component, the isocontours of c′ follow
+markedly different patterns at low and high Pe.
+We now compare these results with those obtained for ordered arrays (black crosses in
+figure 6) and discuss the effect of the microstructure. At small Pe, Dconv
+∥
+and Dconv
+⊥
+grow
+quadratically with Pe in both free and ordered arrays. This scaling was also obtained by
+Koch & Brady (1985) for low-Pe dispersion in porous media with random microstructure
+(albeit in the Stokes flow limit). Since in the low-Pe regime, diffusion by the random
+motion of molecules is much faster than convection by the flow, the microstructure has
+only a quantitative incidence on Dconv, and dispersion is qualitatively identical in ordered
+and freely evolving suspensions. Note that similar features in the spatial distribution of
+c′ can be identified in ordered and free arrays at low Pe (see tubular structures in the
+left side of figures 4 and 7 for Dconv
+∥
+, and quadrupolar ones in the left side of figures 5
+and 8 for Dconv
+⊥
+). We however emphasize that precise quantitative agreement between the
+results for one and for many bubbles at low P´eclet number in figure 6 is not expected, as
+the flows and the microstructures in the two systems are different (Bunner & Tryggvason
+2002; Loisy et al. 2017).
+At high Pe, the Taylor dispersion scaling obtained in ordered arrays is replaced, in both
+directions, by a scaling similar to the one characterizing mechanical dispersion, as soon as
+the relative motion between bubbles is allowed. In this regime, the transverse dispersion
+is indeed governed by mechanical dispersion. Irrespective of the value of Pe, any Taylor
+dispersion in the vertical direction is limited by transverse diffusion or dispersion, the
+latter becoming more significant at large Pe. This results in a scaling similar to that
+of mechanical dispersion in the longitudinal direction as well, such that a distinction
+between these two mechanisms (pure mechanical dispersion, or Taylor dispersion limited
+by transverse mechanical one) cannot be made. Incidentally, mechanical dispersion is also
+obtained at high Pe in random media in Stokes flow conditions (Koch & Brady 1985).
+Although the microstructure of the present bubbly suspensions has not been evaluated
+quantitatively, visual inspection and prior results on their dynamics (Loisy et al. 2017)
+showed that it is not random, but rather characterized by a certain “organization”.
+Despite the fact that freely evolving suspensions resemble ordered ones with respect to
+their dynamics, scalar dispersion is extremely sensitive to the presence of disorder, and is
+fundamentally different in perfectly ordered and weakly disordered suspensions at high
+P´eclet number. It does not, however, seem to be sensitive to the degree of disorder, as
+suggested by the fact that the same scalings with Pe are obtained for random porous
+media and weakly disordered suspensions. We stress that this last statement is purely
+speculative, and would require a quantitative study of the effect of the microstructure to
+be confirmed.
+We finally attempt a comparison of our results with the experimental data of Alm´eras
+et al. (2015), who measured the effective diffusivity of a homogeneous swarm of high-
+Reynolds-number rising bubbles at Pe ≈ 1.75×106 for gas volume fractions ranging from
+1 % to 13 %. It is important to stress that in these experiments, Re ≈ 700, whereas in the
+simulations, Re ≈ 30, so the comparison is only indicative. Interpolation (by eye) of their
+data at φ ≈ 2.4 % (figure 10 in their paper) yields Deff
+∥ /D = 1×106 and Deff
+⊥ /D = 5×105.
+These experimental values are represented by purple stars in figure 6. Note that at
+such high P´eclet number, the dominant contribution to Deff is due to Dconv, so it seems
+reasonable to assume that these are equivalent. The order of magnitude of Deff
+∥ /D is
+comparable in the experiment and in the simulation, whereas Deff
+⊥ /D is much higher
+in the experiment. This difference can be explained from the different properties of the
+
+The effective diffusivity of ordered and freely evolving bubbly suspensions
+19
+numerical and experimental flows considered: partition coefficient (the dye concentration
+in the gas is presumably zero in the experiments from Alm´eras et al. (2015)), diffusivity
+ratio, and bubble-induced liquid agitation in the horizontal direction. In our simulations
+of free arrays at moderate Re, the bubbles were indeed observed to rise along nearly
+straight vertical lines, and the anisotropy ratio characterizing the liquid velocity variance,
+2⟨u′
+3u′
+3⟩/⟨u′
+1u′
+1 + u′
+2u′
+2⟩, is approximately 8 (for Nb = 8), whereas in the experiment
+at high Re, the bubble motion is fully three-dimensional, and the anisotropy ratio is
+approximately 2. Finally, as only one value of the P´eclet number was considered in the
+experiments of Alm´eras et al. (2015), no comparison of their data with our results can
+be offered regarding the dependence of the effective diffusivity on the P´eclet number.
+6. Conclusions
+In this study we investigated scalar dispersion in homogeneous bubbly suspensions as
+described by an effective diffusivity tensor. The longitudinal and transverse components
+of the convective contribution to the effective diffusivity, denoted Dconv
+∥
+and Dconv
+⊥
+,
+respectively, have been computed for bubbly suspensions in various flow regimes. This
+convective contribution is that associated with bubble-induced agitation, and is the
+dominant contribution to the effective diffusivity in commonly encountered bubbly flows.
+The dispersion theory of Koch et al. (1989) indicates that convective mixing mech-
+anisms in ordered suspensions in Stokes-flow conditions differ at low and high P´eclet
+numbers. According to this theory, when the bulk flow is aligned with a primary axis
+of a simple cubic lattice of spheres, convectively enhanced dispersion is expected at low
+P´eclet number, whereas Taylor dispersion should dominate at high P´eclet number. In
+the present study, we have extended this theory to account for weak inertial effects,
+and we have shown that these two dispersion regimes are qualitatively unchanged in the
+presence of (weak) inertia. This result has been confirmed by direct numerical simulations
+for values of the Reynolds number ranging from vanishingly small to moderate. In all
+investigated cases, Dconv
+∥
+was found to be significantly larger than Dconv
+⊥
+, and theoretical
+predictions have been shown to yield the correct qualitative behaviour and order of
+magnitude of both Dconv
+∥
+and Dconv
+⊥
+in a variety of flow regimes (spherical to strongly
+deformed bubbles with Reynolds numbers up to 10) at small volume fraction.
+Direct numerical simulations of scalar transport in freely evolving bubbly suspensions,
+as represented by free arrays of bubbles, have been carried out for a wide range of P´eclet
+numbers, and the effect of introducing additional degrees of freedom in the system has
+been evaluated. At low P´eclet number, dispersion in free arrays is convectively enhanced,
+as in ordered ones. At high P´eclet number, in freely evolving suspensions wherein at
+least two bubbles are present in a unit cell, the longitudinal component of the effective
+diffusivity exhibits a scaling that is similar to that characterizing mechanical dispersion.
+This suggests that the limiting role of molecular diffusion to Taylor dispersion is taken
+over by mechanical dispersion, or that mechanical dispersion itself dominates. Besides,
+the effective diffusivity seems to be weakly sensitive to the number of bubbles present in
+a unit cell. This last assertion requires more thorough investigations to be confirmed,
+but is encouraging regarding the possibility of computing the effective diffusivity of
+homogeneous bubbly flows from direct numerical simulations of systems of relatively
+small size. This would allow in particular a thorough investigation of the roles played by
+the volume fraction and the flow regime, which could not be undertaken as part of the
+present study.
+The results presented in this paper are restricted to bubbles having the same diffusivity
+as that of the surrounding liquid, and to scalar fields that are continuous across the
+
+20
+A. Loisy, A. Naso, P. D. M. Spelt
+interface, and therefore cannot be straightforwardly compared to those obtained in real
+bubbly flows. A jump in the scalar field, which represents the difference in solubilities
+given by Henry’s law in the context of chemical species transport, as well as a difference
+in diffusivities, would introduce a diffusive contribution to the effective diffusivity tensor
+(2.5) in addition to the convective one considered in this study. The present results
+show the convective contribution at large P´eclet numbers and modest volume fraction
+to be substantially larger than the diffusive contribution from nonequal diffusivities or
+solubilities (Maxwell 1873; Jeffrey 1973; Koch & Brady 1985). A difference in diffusivities
+or solubilities would however also have some indirect effect on the convective contribution,
+which magnitude should be investigated in the future.
+Besides the effective diffusivity, another quantity of practical importance is the rate
+of interfacial scalar transport in the presence of an average scalar gradient between the
+disperse phase and the bulk. Heat and mass exchanges across phase boundaries are
+traditionally expressed as dimensionless transfer coefficients called the Nusselt and the
+Sherwood numbers, respectively. Their functional dependences on suspension properties,
+in particular the volume fraction, have been the subject of analytical (Acrivos et al.
+1980), numerical (Aboulhasanzadeh & Tryggvason 2014), and experimental (Colombet
+et al. 2011, 2015) studies. Formally, the Nusselt and the Sherwood numbers are closure
+coefficients for the conditionally averaged scalar transport equation, where the conditional
+average is defined as an ensemble average over the subset of realizations wherein a
+particulate is present at a given position. Less formally, the Nusselt and Sherwood
+numbers are related to a “mesoscale” description of scalar transfer between the two
+phases, whereas the effective diffusivity is associated with a “macroscale” description of
+scalar transport through a two-phase mixture seen as a continuum. They correspond to
+different closure problems, and one cannot be inferred from the other. Nevertheless, the
+present work will be primarily important for mass transfer processes in bubbly flows that
+are liquid-phase controlled. This is because then the mixture concentration distribution
+is key, whereas if it is gas-phase controlled, the concentration in the liquid will be almost
+uniform and one is primarily concerned by the circumstances inside each bubble.
+This work benefited from the financial support of the French research agency (grant
+ANR-12-BS09-0011), and was performed using the HPC resources provided by GENCI-
+CINES and GENCI-IDRIS (grant x20162b6893), PSMN (´Ecole Normale Sup´erieure de
+Lyon), P2CHPD (Universit´e Claude Bernard Lyon 1) and PMCS2I (´Ecole Centrale de
+Lyon).
+Appendix: Spatial convergence tests
+We present the results of some spatial convergence tests of the algorithm solving the
+scalar transport equation. The results of similar tests for the algorithm solving the flow
+are shown in Loisy et al. (2017); Loisy (2016).
+The effect of the grid spacing on Dconv
+∥
+and Dconv
+⊥
+has been assessed for case E1 at
+Pe = 103 for one value of the volume fraction (φ = 2.4 %), in both ordered and
+free configurations. For ordered arrays, three different resolutions were tested, namely
+db/∆x = {20, 40, 60} with ∆x the grid spacing. The results are shown in figure 9. The
+error in the values of Dconv
+∥
+and Dconv
+⊥
+arising from spatial discretization is less than 1 %
+when a resolution of 40 grid cells per bubble diameter is used. This resolution is the same
+as that used for the simulation of the corresponding bubbly flow in Loisy et al. (2017).
+In practice, we used for each configuration the same resolution as that selected for the
+simulation of the corresponding ordered bubbly suspensions (see Loisy et al. (2017)),
+
+The effective diffusivity of ordered and freely evolving bubbly suspensions
+21
+log10 (∆x db)
+−2
+−1.8
+−1.6
+−1.4
+−1.2
+−1
+−4
+−3
+−2
+−1
+0
+log10 
+�
+�
+�
+Dconv − D∆x=0
+conv
+D∆x=0
+conv
+�
+�
+�
+n = 3
+n = 4
+longitudinal component
+transverse component
+Figure 9. Spatial convergence for an ordered array of bubbles in case E1 at Pe = 103: relative
+error in Dconv
+∥
+and Dconv
+⊥
+as a function of the grid spacing ∆x (db is the bubble volume-equivalent
+diameter; Dconv
+∆x=0 is extrapolated assuming Dconv = Dconv
+∆x=0 − k∆xn, where the values of the
+three parameters Dconv
+∆x=0, k and n are fitted from numerical data).
+namely 60 grid cells per diameter for case C and 40 grid cells per diameter for the other
+cases.
+For free arrays, due to the computational cost of the simulations, only two different
+resolutions were tested, namely 20 and 30 grid cells per bubble diameter, for an array
+of 8 bubbles. Simulations at higher resolution were too expensive to be continued over
+sufficiently long times to allow a quantitative estimate of the uncertainty. However the
+values of Dconv
+∥
+and Dconv
+⊥
+obtained with the finer grid, depicted by filled red squares
+in figure 6, are nearly indistinguishable from those obtained with the coarser grid. A
+resolution of 20 grid cells per diameter was therefore concluded to be sufficient for free
+arrays in view of the present purposes.
+For a given case, the same resolution was used for all P´eclet numbers. Note that when
+the gas diffusivity differs from that of the liquid (a situation not considered here but
+frequently encountered in practice), finer resolutions may be required, as thin scalar
+boundary layers around the bubbles would then need to be resolved.
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+Society s2-20 (1), 196–212.
+

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+Integrable systems with linear periodic integral
+for the Lie algebra e(3)
+I. K. Kozlov∗
+and
+A. A. Oshemkov†
+Abstract
+Integrable systems with a linear periodic integral for the Lie algebra e(3) are
+considered. One investigates singulariries of the Liouville foliation, bifurcation
+diagram of the momentum mapping, transformations of Liouville tori, topology
+of isoenergy surfaces and other topological properties of such systems.
+Keywords and phrases: Integrable Hamiltonian system, periodic integral,
+bifurcation diagram, momentum mapping, Liouville tori
+1
+Introduction
+In this paper we study some topological properties of integrable Hamiltonian
+systems with an 𝑆1-symmetry given by the Euler equations for the Lie algebra
+e(3). Probably, the most well-known example of such a system is the classical
+Lagrange top. Roughly speaking, we consider a “generalized” Lagrange top which
+Hamiltonian has an arbitrary potential function and linear terms in momenta,
+but possesses the same 𝑆1-symmetry.
+We are interested in local and global topological properties of the Liouville
+foliation defined by the system under consideration, namely, the structure of bi-
+furcation diagram and transformations of Liouville tori for critical values of the
+momentum mapping, non-degeneracy of equilibria and other singular points, the
+topology of isoenergy surfaces.
+Note that there is a number of integrable systems with periodic linear inte-
+gral which are well known in mechanics and mathematical physics, which phase
+topology were studied by various authors. In particular, there are Lagrange and
+Kirchhoff integrable cases in rigid body dynamics (for the description of their
+topology see [1–3]), the integrable case of Leggett equations describing dynamics
+of spin in the superfluid 3He (the bifurcation diagram and Fomenko invariants
+for this system are described in [6]), the integrable case of the motion of heavy
+ellipsoid on a smooth horizontal plane (topological invariants for this system were
+found in [7]).
+∗No Affiliation, E-mail: [email protected]
+†Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991 Russia, E-mail:
+[email protected]
+3
+arXiv:2301.05283v1  [math.DG]  12 Jan 2023
+
+Topological properties of all these systems are quite similar because of an 𝑆1-
+symmetry which imposes strong restrictions on the structure of their singularities.
+Therefore, they can be studied under a uniform scheme. In this paper we perform
+such an investigation for an example of Hamiltonian possessing a periodic linear
+integral on e(3)*. Note that the problem of topological investigation of integrable
+systems with S1-action is discussed in paper [4], which contains a list of various
+open problems in the theory of integrable systems.
+Apart from the systems on e(3)* considered in this paper there are other
+integrable systems with 𝑆1-symmetry, which were also studied by various authors.
+For instance, natural mechanical systems on surfaces of revolution homeomorphic
+to the sphere were studied recently in [5] (see also [1]). Another example is the
+classical Euler case in the rigid body dynamics, where the 𝑆1-action is given not
+by a linear, but by a quadratic integral. The results obtained in this paper show
+in particular that there are some differences between the topological properties
+of the systems under consideration and other cases with an 𝑆1-symmetry (for
+example, the one investigated in [5] or the Euler case).
+The article is organized as follows. In Section 2 we describe the systems under
+consideration. We start the analysis with the study of non-deneracy and types of
+singular points of rank 0 in Section 3 (Corollary 1 ). In Section 4 we find singular
+points of rank 1 (Theorem 3) and describe the bifurcation diagrams of the system
+(Theorems 4 and 5). In Section 5 we determine types of non-degenerate points
+of rank 1 (Theorem 6) and specify the corresponding Liouville tori bifurcations
+(Theorem 7). Finally, in Section 6 we list all possible isoenergy surfaces for the
+system (Theorem 8).
+2
+Description of the system
+Let us recall that the Lie–Poisson bracket for the Lie algebra e(3) is given by the
+formulas
+{𝑆𝑖, 𝑆𝑗} = 𝜀𝑖𝑗𝑘𝑆𝑘,
+{𝑆𝑖, 𝑅𝑗} = 𝜀𝑖𝑗𝑘𝑅𝑘,
+{𝑅𝑖, 𝑅𝑗} = 0,
+(1)
+where 𝑆1, 𝑆2, 𝑆3, 𝑅1, 𝑅2, 𝑅3 are linear coordinates on the dual space e(3)* for the
+Lie algebra e(3). We will use the notation S = (𝑆1, 𝑆2, 𝑆3) and R = (𝑅1, 𝑅2, 𝑅3)
+and also ⟨·,·⟩ and × for the scalar and vector product of 3-dimensional vectors.
+A Hamiltonian system with Hamiltonian 𝐻 is given by the Euler equations
+˙𝑥𝑖 = {𝑥𝑖, 𝐻},
+which for the Lie algebra e(3) take the form
+˙S = 𝜕𝐻
+𝜕S × S + 𝜕𝐻
+𝜕R × R,
+˙R = 𝜕𝐻
+𝜕S × R.
+Bracket (1) has two Casimir functions:
+𝐹1 = ⟨R, R⟩,
+𝐹2 = ⟨S, R⟩.
+Their regular common level surfaces
+𝑀4
+𝑎,𝑔 = {(S, R) | 𝐹1(S, R) = 𝑎, 𝐹2(S, R) = 𝑔, },
+𝑎 > 0,
+(2)
+4
+
+are the sympectic leaves of bracket (1) and are the orbits of the coadjoint repsre-
+sentation for the Lie algebra e(3). We are interested in integrable Hamiltonian
+systems on the orbits 𝑀4
+𝑎,𝑔 for which some linear function on e(3)* is a first integral
+defining an 𝑆1-action.
+Let us describe several examples of such systems from mechanics and math-
+ematical physics, which are integrable cases of the Euler equations for the Lie
+algebra e(3) with Hamiltonian 𝐻 and integral 𝐾 (an explanation of physical sense
+for parameters and variables of these systems can be found in [1,2,6,7]).
+1) The Lagrange case. This is a symmetric top with two equal moments of
+inertia which center of gravity lies on the symmetry axis:
+𝐻 = 𝑆2
+1
+𝐴 + 𝑆2
+2
+𝐴 + 𝑆2
+3
+𝐵 − 𝑝𝑅3,
+𝐾 = 𝑆3,
+where 𝐴, 𝐵, 𝑝 = const.
+2) The Kirchhoff case. This system describes the motion of a dynamically
+symmetric rigid body in an ideal fluid:
+𝐻 = 𝐴𝑆2
+1 + 𝐴𝑆2
+2 + 𝑎𝑆2
+3 + 2(𝐵𝑆1𝑅1 + 2𝐵𝑆2𝑅2 + 𝑏𝑆3𝑅3)+
++ 𝐶𝑅2
+1 + 𝐶𝑅2
+2 + 𝑐𝑅2
+3,
+𝐾 = 𝑆3,
+where 𝐴, 𝑎, 𝐵, 𝑏, 𝐶, 𝑐 = const.
+3) The following integrable case for the Leggett system describing the dynamics
+of spin in the superfluid 3He:
+𝐻 = 𝑆2
+1 + 𝑆2
+2 + 𝑆2
+3 − 𝛾𝑆3 − 𝑅2
+3,
+𝐾 = 𝑆3,
+where 𝛾 = const.
+4) Integrable system describing the motion of a dynamically and geometrically
+symmetric heavy ellipsoid on a smooth horizontal plane:
+𝐻 = 𝑆2
+1 + 𝑆2
+2 + 𝐴(𝑆1𝑅1 + 𝑆2𝑅2)2
+2𝑏(1 + 𝐴(𝑅2
+1 + 𝑅2
+2))
++ 𝑆2
+3
+2𝐽 +
+√︁
+1 + 𝑐𝑅2
+3 + 𝑠𝑅3,
+𝐾 = 𝑆3
+where 𝐴 =
+𝑐𝑅2
+3
+1 + 𝑐𝑅2
+3
+,
+𝑏, 𝑐, 𝐽, 𝑠 = const.
+In all these examples the additional integral is the function 𝑆3 on e(3)*. Let
+us explain that this is a general case if we require that the integral is linear and
+periodic.
+Assertion 1. Let 𝐾 be a linear functions on e(3)* which Hamiltonian flow sgrad 𝐾
+defined by bracket (1) is periodic. Then there is a linear change of variables pre-
+serving the bracket (1) taking the function 𝐾 to 𝑐𝑆3, where 𝑐 is some constant.
+Proof. Let 𝐾 = 𝛼1𝑆1 + 𝛼2𝑆2 + 𝛼3𝑆3 + 𝛽1𝑅1 + 𝛽2𝑅2 + 𝛽3𝑅3. For an arbitrary or-
+thogonal matrix 𝐴 the transformation Φ𝐴 : (S, R) → (𝐴S, 𝐴R) preserves bracket
+(1).
+If 𝛼1 = 𝛼2 = 𝛼3 = 0, then we can choose a matrix 𝐴 such that Φ𝐴
+takes the function 𝐾 to 𝜆𝑅3, where 𝜆 = const. It is clear that the Hamilto-
+nian flow of the function 𝜆𝑅3 is not periodic, since the trajectories of the field
+sgrad 𝑅3 = (−𝑅2, 𝑅1, 0, 0, 0, 0) are straight lines in e(3)*.
+If there are non-zero 𝛼𝑖, then applying an appropriate transformation Φ𝐴 we
+can transform 𝐾 to a function of the form 𝑐𝑆3 + 𝛽′
+1𝑅1 + 𝛽′
+2𝑅2 + 𝛽′
+3𝑅3. It is easy
+to check that for any vector v the transformations Ψv : (S, R) → (S + v × R, R)
+5
+
+also preserve bracket (1). This allows one to transform the function 𝐾 to the form
+𝑐𝑆3 + 𝜆𝑅3, where 𝑐 ̸= 0.
+Now consider the function 𝐾 = 𝑆3 + 𝜆𝑅3 and determine for which 𝜆 the
+Hamiltonian flow of 𝐾 is periodic. Integral trajectories for the field sgrad 𝐾 =
+(−𝑆2 − 𝜆𝑅2, 𝑆1 + 𝜆𝑅1, 0, −𝑅2, 𝑅1, 0) can be explicitly written:
+𝛾(𝑡) = ((𝑠1−𝜆𝑟2𝑡) cos 𝑡−(𝑠2+𝜆𝑟1𝑡) sin 𝑡, (𝑠2+𝜆𝑟1𝑡) cos 𝑡+(𝑠1−𝜆𝑟2𝑡) sin 𝑡,
+𝑠3, 𝑟1 cos 𝑡 − 𝑟2 sin 𝑡, 𝑟2 cos 𝑡 + 𝑟1 sin 𝑡, 𝑟3),
+where 𝑠1, 𝑠2, 𝑠3, 𝑟1, 𝑟2, 𝑟3 are constants.
+It is clear from this formula that the
+trajectories are periodic only for 𝜆 = 0.
+Remark 1. It is well known that an action of any compact group can be linearized
+at a fixed point and that for an action of the circle 𝑆1 the corresponding tangent
+space can be represented as a sum of invariant two-dimensional subspaces. Thus
+among all linear functions on e(3)* the periodic integrals are distiguished by the
+property that their linearization at any singular point is a unitary operator with
+respect to a complex structure on the tangent space. It also follows that up to
+the choice of the coordinate system and multipltication by a constant any periodic
+linear integral on e(3)* is 𝑆3.
+Further we will consider Hamiltonian systems for the Lie algebra e(3) which
+possess the first integral 𝐾 = 𝑆3 and which Hamiltonian 𝐻 is quadratic in 𝑆, i.e.,
+𝐻 = 𝐴1𝑆2
+1 + 𝐴2𝑆2
+2 + 𝐴3𝑆2
+3 + 𝑓1(R)𝑆1 + 𝑓2(R)𝑆2 + 𝑓3(R)𝑆3 + 𝑓4(R),
+(3)
+where 𝐴1, 𝐴2, 𝐴3 are arbitrary positive constants and 𝑓1, 𝑓2, 𝑓3, 𝑓4 are smooth
+functions of 𝑅1, 𝑅2, 𝑅3.
+First of all, let us rewrite Hamiltonian (3) in a more convient way using its
+commutativity with the function 𝑆3.
+Assertion 2. Up to multiplication by a constant any Hamiltonian of the form (3)
+commuting with the function 𝐾 = 𝑆3 has the form
+𝐻 = 1
+2
+(︁
+𝑆2
+1 + 𝑆2
+2 + 𝑆2
+3
+𝛽
+)︁
++ 𝑔1(R2, 𝑅3)(𝑆1𝑅2 − 𝑆2𝑅1)+
++ 𝑔2(R2, 𝑅3)⟨S, R⟩ + 𝑔3(R2, 𝑅3)𝑆3 + 𝑉 (R2, 𝑅3),
+(4)
+where 𝛽 > 0 and the functions 𝑔1, 𝑔2, 𝑔3, 𝑉 depend only on R2 and 𝑅3 and are
+smooth if R2 ̸= 0.
+Proof. The Hamiltonian vector field for the function 𝐾 is equal to
+sgrad 𝐾 = −𝑅2
+𝜕
+𝜕𝑅1
++ 𝑅1
+𝜕
+𝜕𝑅2
+− 𝑆2
+𝜕
+𝜕𝑆1
++ 𝑆1
+𝜕
+𝜕𝑆2
+.
+Since {𝐻, 𝐾} = (sgrad 𝐾)𝐻 = 0, we get
+(sgrad 𝐾)𝐻 = 2(𝐴2 − 𝐴1)𝑆1𝑆2+
++
+(︁
+−𝑅2
+𝜕𝑓1
+𝜕𝑅1
++𝑅1
+𝜕𝑓1
+𝜕𝑅2
++𝑓2(R)
+)︁
+𝑆1 +
+(︁
+−𝑅2
+𝜕𝑓2
+𝜕𝑅1
++𝑅1
+𝜕𝑓2
+𝜕𝑅2
+−𝑓1(R)
+)︁
+𝑆2+
++
+(︁
+−𝑅2
+𝜕𝑓3
+𝜕𝑅1
++ 𝑅1
+𝜕𝑓3
+𝜕𝑅2
+)︁
+𝑆3 +
+(︁
+−𝑅2
+𝜕𝑓4
+𝜕𝑅1
++ 𝑅1
+𝜕𝑓4
+𝜕𝑅2
+)︁
+= 0.
+6
+
+Hence, 𝐴1 = 𝐴2 (multiplying by a constant we can make both these constants
+equal to 1
+2) and the four expressions in the brackets are equal to zero.
+In polar coordinates (𝜌, 𝜙) on the plane (𝑅1, 𝑅2) the vector field
+𝜕
+𝜕𝜙 is exactly
+−𝑅2
+𝜕
+𝜕𝑅1 + 𝑅1
+𝜕
+𝜕𝑅2 . Therefore,
+𝜕𝑓3
+𝜕𝜙 = 0,
+𝜕𝑓4
+𝜕𝜙 = 0,
+𝜕𝑓1
+𝜕𝜙 = −𝑓2,
+𝜕𝑓2
+𝜕𝜙 = 𝑓1.
+The first two of these equations imply that 𝑓3 and 𝑓4 depend only on 𝜌 and
+𝑅3 or, equivalently, 𝑓3(R) = 𝑔3(R2, 𝑅3) and 𝑓4(R) = 𝑉 (R2, 𝑅3). The latter two
+equations can be cosidered as a system of ODE with parameters 𝜌 and 𝑅3. Solving
+it, we obtain
+𝑓1 = 𝑓11(𝜌, 𝑅3) cos 𝜙 + 𝑓12(𝜌, 𝑅3) sin 𝜙 = 𝑓11(𝜌, 𝑅3)
+𝜌
+𝑅1 + 𝑓12(𝜌, 𝑅3)
+𝜌
+𝑅2,
+𝑓2 = −𝑓12(𝜌, 𝑅3) cos 𝜙+𝑓11(𝜌, 𝑅3) sin 𝜙 = −𝑓12(𝜌, 𝑅3)
+𝜌
+𝑅1+𝑓11(𝜌, 𝑅3)
+𝜌
+𝑅2.
+Since 𝜌 =
+√︀
+𝑅2
+1 + 𝑅2
+2 we get the desired form for the Hamiltonian 𝐻.
+3
+Singularities of rank 0
+It turns out that equilibria points for a Hamiltonian system on e(3)* possessing
+a linear periodic integral 𝐾 are exactly the points where sgrad 𝐾 = 0.
+This
+gives the following simple description for singularities of rank 0 of such integrable
+Hamiltonian systems (not necessarily with Hamiltonian of the form (3)).
+Theorem 1. The set of singular points of rank 0 for an integrable Hamiltonian
+system on e(3)* with arbitrary Hamiltonian 𝐻 possessing the integral 𝐾 = 𝑆3 is
+the two-dimensional subspace
+{(0, 0, 𝑆3, 0, 0, 𝑅3)}
+(5)
+in e(3)*. In particular, for each orbit 𝑀4
+𝑎,𝑔 there are precisely two singular points
+of rank 0:
+(︁
+0, 0, ± 𝑔
+√𝑎, 0, 0, ±√𝑎
+)︁
+.
+Proof. The Hamiltonian vector field of a function 𝑓 on e(3)* has the form
+sgrad 𝑓 =
+(︁𝜕𝑓
+𝜕S × S + 𝜕𝑓
+𝜕R × R, 𝜕𝑓
+𝜕S × R
+)︁
+,
+(6)
+and for the function 𝐾 = 𝑆3 we have sgrad 𝐾 = (−𝑆2, 𝑆1, 0, −𝑅2, 𝑅1, 0). There-
+fore, sgrad 𝐾 = 0 exactly at points (5). Thus, points other than (5) can not be
+singular points of rank 0.
+Let us prove that sgrad 𝐻 vanishes at points (5). The functions 𝐻 and 𝐾
+commute with respect to bracket (1), i.e., 𝑑𝑦𝐻(sgrad𝑦 𝐾) = 0 for any point 𝑦 ∈
+e(3)* (the index 𝑦 in 𝑑𝑦𝑓 or sgrad𝑦 𝑓 denotes the point at which the differential
+7
+
+or, respectively, skew-gradient of the function 𝑓 is taken). Taking the differential
+of the function 𝑑𝑦𝐻(sgrad𝑦 𝐾) at any point 𝑦 = (0, 0, 𝑆3, 0, 0, 𝑅3), we get
+𝐴*
+𝐾(𝑑𝑦𝐻) = 0,
+(7)
+where 𝐴𝐾 is the linearization operator for the vector field sgrad 𝐾 at the point 𝑦,
+since sgrad𝑦 𝐾 = 0. The matrix of the operator 𝐴𝐾 : e(3)* → e(3)* has the form
+⎛
+⎜
+⎜
+⎜
+⎜
+⎜
+⎜
+⎝
+0
+−1
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+−1
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+0
+0
+0
+⎞
+⎟
+⎟
+⎟
+⎟
+⎟
+⎟
+⎠
+and therefore condition (7) implies that 𝜕𝐻
+𝜕𝑆1 = 𝜕𝐻
+𝜕𝑆2 = 𝜕𝐻
+𝜕𝑅1 = 𝜕𝐻
+𝜕𝑅2 = 0 at any point
+𝑦 = (0, 0, 𝑆3, 0, 0, 𝑅3).
+Hence sgrad 𝐻 vanishes at points (5), since at a point
+𝑦 = (0, 0, 𝑆3, 0, 0, 𝑅3) formula (6) becomes
+sgrad𝑦 𝑓 =
+(︁
+𝑆3
+𝜕𝑓
+𝜕𝑆2
++ 𝑅3
+𝜕𝑓
+𝜕𝑅2
+, −𝑆3
+𝜕𝑓
+𝜕𝑆1
+− 𝑅3
+𝜕𝑓
+𝜕𝑅1
+, 0, 𝑅3
+𝜕𝑓
+𝜕𝑆2
+, −𝑅3
+𝜕𝑓
+𝜕𝑆1
+, 0
+)︁
+.
+Theorem 1 is proved.
+Now, let us state when these zero-rank points are non-degenerate and deter-
+mine their type (for more information about non-degeneracy of singular points of
+a momentum mapping see [1]).
+Theorem 2. For an integrable Hamiltonian system on e(3)* with arbitrary Hamil-
+tonian 𝐻 possessing the integral 𝐾 = 𝑆3, the singular point of rank 0
+𝑃± =
+(︁
+0, 0, ± 𝑔
+√𝑎, 0, 0, ±√𝑎
+)︁
+on the orbit 𝑀4
+𝑎,𝑔 is non-degenerate iff 𝑞 ̸= 0, where
+𝑞 = 𝑝2 + 𝑅2
+3(𝐻11𝐻22 − |𝐻12|2),
+(8)
+𝑝 =
+𝑔
+2𝑅3
+𝜕2𝐻
+𝜕𝑆2
+1
++ 𝑅3
+𝜕2𝐻
+𝜕𝑆1𝜕𝑅1
+− 𝜕𝐻
+𝜕𝑆3
+,
+(9)
+and
+𝐻11 = 𝜕2𝐻
+𝜕𝑆2
+1
+,
+𝐻12 =
+(︁
+𝜕2𝐻
+𝜕𝑆1𝜕𝑅1
+− 1
+𝑅3
+𝜕𝐻
+𝜕𝑆3
+)︁
++ 𝑖
+𝜕2𝐻
+𝜕𝑆2𝜕𝑅1
+,
+𝐻22 = 𝜕2𝐻
+𝜕𝑅2
+1
++ 𝑔
+𝑅3
+3
+𝜕𝐻
+𝜕𝑆3
+− 1
+𝑅3
+𝜕𝐻
+𝜕𝑅3
+.
+Also, if the point 𝑃± is non-degenerate, then its type is
+1. center-center if 𝑞 > 0,
+2. focus-focus if 𝑞 < 0.
+Theorem 2 holds for any Hamiltonian 𝐻 that commutes (and is functionally
+independent) with 𝐾 = 𝑆3. For the Hamiltonian 𝐻 quadratic in S the condition
+of non-degeneracy and types of singular points of rank 0 are as follows.
+8
+
+Corollary 1. For Hamiltonian (4) the type of singular points of rank 0 is com-
+pletely determined as in Theorem 2 by
+𝑞 = 𝑔2
+4𝑅2
+3
+− 𝑅2
+3𝑔2
+1(𝑎, 𝑅3) + 𝑔𝑅3
+𝜕𝑔2
+𝜕𝑅3
+(𝑎, 𝑅3) − 𝑔 𝜕𝑔3
+𝜕𝑅3
+(𝑎, 𝑅3) − 𝑅3
+𝜕𝑉
+𝜕𝑅3
+(𝑎, 𝑅3).
+Proof. Calculating all expressions from Theorem 2, we have
+𝐻11 = 1,
+𝐻12 = − 1
+𝑅3
+(︁ 𝑔
+𝛽𝑅3
++ 𝑔3(𝑎, 𝑅3)
+)︁
+− 𝑖𝑔1(𝑎, 𝑅3),
+𝐻22 = 𝑔
+𝑅3
+3
+(︁ 𝑔
+𝛽𝑅3
++ 𝑔3(𝑎, 𝑅3)
+)︁
+−
+− 1
+𝑅3
+(︁
+𝑔 𝜕𝑔2
+𝜕𝑅3
+(𝑎, 𝑅3) + 𝑔
+𝑅3
+𝜕𝑔3
+𝜕𝑅3
+(𝑎, 𝑅3) + 𝜕𝑉
+𝜕𝑅3
+(𝑎, 𝑅3)
+)︁
+,
+(10)
+and
+𝑝 = 𝑔
+𝑅3
+(︁1
+2 − 1
+𝛽
+)︁
+− 𝑔3(𝑎, 𝑅3).
+Substituting them into (8), one obtains the required formula for 𝑞.
+In order to prove Theorem 2 we use the following criteria of non-degeneracy
+(see [1]), which can be regarded as a definition.
+Definition 1. A point 𝑃 of rank 0 for an integrable Hamiltonian system with
+Hamiltonian 𝐻 and integral 𝐾 on a symplectic manifold 𝑀4 is non-degenerate iff
+the following two conditions hold:
+• the linearizations 𝐴𝐻 and 𝐴𝐾 of the Hamiltonian vector fields sgrad 𝐻 and
+sgrad 𝐾 at the point 𝑃 are linear independent,
+• there exists a linear combination 𝜆𝐴𝐻 + 𝜇𝐴𝐾 with four different non-zero
+eigenvalues.
+Let us study the spectrum of linearization of sgrad 𝐻 at the points of rank 0.
+Taking functions 𝑆1, 𝑆2, 𝑅1, 𝑅2 as local coordinates in a neighbourhood of 0-rank
+point 𝑃± on an orbit 𝑀4
+𝑎,𝑔 we have
+𝑅3 = ±
+√︁
+𝑎 − 𝑅2
+1 − 𝑅2
+2,
+𝑆3 = 1
+𝑅3
+(𝑔 − 𝑆1𝑅1 − 𝑆2𝑅2).
+Denote by ̂︀𝐻(𝑆1, 𝑆2, 𝑅1, 𝑅2) the restriction of the fucntion 𝐻 onto 𝑀4
+𝑎,𝑔.
+Lemma 1. For any function 𝐻 commuting with 𝐾 = 𝑆3 the spectrum of the
+linearization operator 𝐴 ̂︀
+𝐻 = Lin(sgrad ̂︀𝐻) at the singular points 𝑃± of rank 0 has
+the form 𝜎(𝐴 ̂︀
+𝐻) = {±𝑖(𝑝 + √𝑞), ±𝑖(𝑝 − √𝑞)}, where 𝑝 and 𝑞 are given by (9) and
+(8).
+Proof. In the coordinates 𝑆1, 𝑆2, 𝑅1, 𝑅2 the Poisson bracket on the symplectic leaf
+𝑀4
+𝑎,𝑔 has the form
+𝒜 =
+⎛
+⎜
+⎜
+⎝
+0
+𝑆3
+0
+𝑅3
+−𝑆3
+0
+−𝑅3
+0
+0
+𝑅3
+0
+0
+−𝑅3
+0
+0
+0
+⎞
+⎟
+⎟
+⎠ .
+9
+
+It is easy to check that the linearization of sgrad 𝐾 defines a complex structure
+on the tangent space:
+𝐴 ̂︀
+𝐾 = Lin(sgrad ̂︀𝐾) =
+⎛
+⎜
+⎜
+⎝
+0
+−1
+0
+0
+1
+0
+0
+0
+0
+0
+0
+−1
+0
+0
+1
+0
+⎞
+⎟
+⎟
+⎠ .
+(11)
+Since [𝐴 ̂︀
+𝐻, 𝐴 ̂︀
+𝐾] = 0, the operator 𝐴 ̂︀
+𝐻 can be complexified. The matrix of
+the Poisson structure can also be complexified, i.e., we can identify (2 × 2)-blocks
+(︁
+𝛼 −𝛽
+𝛽
+𝛼
+)︁
+in matrices with complex numbers 𝛼 + 𝑖𝛽. Thus, in the complex coordi-
+nates 𝑆1 + 𝑖𝑆2, 𝑅1 + 𝑖𝑅2 the matrix 𝒜 of the Poisson structure has the form
+𝒜 =
+(︂−𝑖𝑆3
+−𝑖𝑅3
+−𝑖𝑅3
+0
+)︂
+.
+On a symplectic manifold we have 𝐴 ̂︀
+𝐻 = 𝒜 𝑑2 ̂︀𝐻, and therefore 𝑑2 ̂︀𝐻 can also be
+complexified. By direct calculation we get
+𝑑2 ̂︀𝐻 =
+(︂𝐻11
+𝐻12
+𝐻12
+𝐻22
+)︂
+,
+where 𝐻𝑙𝑗 are given by formulas (10). The imaginary parts of 𝐻11 and 𝐻22 vanish
+because 𝐻 commutes with 𝐾.
+Using the fact that if 𝜇1, 𝜇2 are eigenvalues of a matrix (𝐴 + 𝑖𝐵) for real ma-
+trices 𝐴, 𝐵, then the matrix
+(︀ 𝐴
+𝐵
+−𝐵 𝐴
+)︀
+has the eigenvalues 𝜇1, 𝜇2, 𝜇1, 𝜇2, we obtain
+that the specturm of the (real) operator 𝐴 ̂︀
+𝐻 is given by the equation
+𝜇2 − 𝑖(𝑆3𝐻11 + 𝑅3𝐻12 + 𝑅3𝐻12)𝜇 + 𝑅2
+3(𝐻11𝐻22 − |𝐻12|2) = 0,
+which solutions give the desired spectrum. Lemma 1 is proved.
+Remark 2. It is clear from (11) that for the integral 𝐾 = 𝑆3 the spectrum of the
+corresponding operator 𝐴 ̂︀
+𝐾 is 𝜎(𝐴 ̂︀
+𝐾) = {𝑖, −𝑖, 𝑖, −𝑖}. This doesn’t immediately
+prove non-deneracy of points but shows that non-degenerate points can be only of
+center-center or focus-focus type.
+Proof of Theorem 2. Using Lemma 1 and Definition 1 of non-degeneracy we get
+the condition of the theorem in all cases except for 𝑞 = 0 or 𝑝2 = 𝑞.
+If 𝑞 = 0, then the spectra of 𝐴 ̂︀
+𝐻 and 𝐴 ̂︀
+𝐾 are proportional, thus the point is
+degenerate (this is precisely the moment when the image of a focus-focus point
+meets an arc of the bifurcation diagram while transforming into a center-center
+point).
+If 𝑝2 = 𝑞, then the point is non-degenerate, and one should just take another
+linear combination with different eigenvalues (such a linear combination exists
+since the spectra of 𝐴 ̂︀
+𝐻 and 𝐴 ̂︀
+𝐾 are non-proportional).
+10
+
+4
+Bifurcation diagrams
+In order to construct the bifurcation diagram let us describe all critical points of
+the momentum mapping. The singular points of rank 0 are found in Section 3.
+Thus, it remains to describe only singular points of rank 1. The next two lemmas
+show that we can use some convenient coordinates for investigating them.
+Lemma 2. For a Hamiltonian system with Hamiltonian 𝐻 of the form (4) and
+integral 𝐾 = 𝑆3, the subspace {(S, R) | 𝑅1 = 𝑅2 = 0} in e(3)* does not contain
+points of rank 1.
+Proof. Since we know all singular points of rank 0 (they are points with 𝑅1 = 𝑅2 =
+𝑆1 = 𝑆2 = 0; see Theorem 1), it suffices to prove that if 𝑦 = (𝑆1, 𝑆2, 𝑆3, 0, 0, 𝑅3) ∈
+e(3)* is a singular point, then its coordinates 𝑆1 and 𝑆2 vanish. Suppose that
+this is not the case.
+Then sgrad𝑦 𝐾 = (−𝑆2, 𝑆1, 0, 0, 0, 0) ̸= 0 and, therefore,
+sgrad𝑦 𝐻 = 𝜆 sgrad𝑦 𝐾 for a certain 𝜆. Hence, by formula (6) (taking into account
+that 𝑅3 ̸= 0), we have 𝜕𝐻
+𝜕𝑆1 = 𝜕𝐻
+𝜕𝑆2 = 0 at the point 𝑦. But for a Hamiltonian of
+the form (4) this is possible only if 𝑆1 = 𝑆2 = 0 for the point 𝑦.
+Now, since we can assume that 𝑅2
+1 + 𝑅2
+2 ̸= 0, we choose new coordinates on
+the remaining set of points 𝑈 = R6(S, R) ∖ {𝑅1 = 𝑅2 = 0}. Note that the set 𝑈
+is homeomorphic to R5 × 𝑆1.
+Lemma 3. Formulas
+𝑆1 = (𝑔 − 𝑘𝑥) cos 𝜙 + 𝑚 sin 𝜙
+√
+𝑎 − 𝑥2
+,
+𝑆2 = (𝑔 − 𝑘𝑥) sin 𝜙 − 𝑚 cos 𝜙
+√
+𝑎 − 𝑥2
+,
+𝑆3 = 𝑘,
+𝑅1 =
+√︀
+𝑎 − 𝑥2 cos 𝜙,
+𝑅2 =
+√︀
+𝑎 − 𝑥2 sin 𝜙,
+𝑅3 = 𝑥
+(12)
+define regular coordinates (𝑥, 𝑚, 𝜙, 𝑘, 𝑎, 𝑔) on the set 𝑈, where 𝑥2 < 𝑎 and 𝜙 is an
+angular coordinate, i.e., is defined modulo 2𝜋.
+The inverse change of variables on the set 𝑈, i.e., the expression of (𝑥, 𝑚, 𝜙, 𝑘, 𝑎, 𝑔)
+through (S, R) is as follows:
+𝑥 = 𝑅3,
+𝑚 = 𝑀(S, R) = 𝑆1𝑅2 − 𝑆2𝑅1,
+𝜙 = arg(𝑅1 + 𝑖𝑅2),
+𝑘 = 𝑆3,
+𝑎 = 𝐹1(S, R) = ⟨R, R⟩,
+𝑔 = 𝐹2(S, R) = ⟨S, R⟩.
+Proof. By direct calculation, it is easy to check that given formulas define a bi-
+jection and that the Jacobian does not vanish on 𝑈:
+det
+𝜕(𝑥, 𝑚, 𝜙, 𝑘, 𝑎, 𝑔)
+𝜕(𝑆1, 𝑆2, 𝑆3, 𝑅1, 𝑅2, 𝑅3) = 2(𝑅2
+1 + 𝑅2
+2) ̸= 0.
+Substituting expressions (12) into (4), we obtain that the Hamiltonian in the
+coordinates (𝑥, 𝑚, 𝜙, 𝑘, 𝑎, 𝑔) on the set 𝑈 has the form
+𝐻 = (𝑔−𝑘𝑥)2+𝑚2
+2(𝑎 − 𝑥2)
++ 𝑘2
+2𝛽 + 𝑔1(𝑎, 𝑥)𝑚 + 𝑔2(𝑎, 𝑥)𝑔 + 𝑔3(𝑎, 𝑥)𝑘 + 𝑉 (𝑎, 𝑥).
+(13)
+Futher we will often write 𝑔1, 𝑔2, 𝑔3, 𝑉 without arguments assuming that they
+are functions of 𝑎 and 𝑥.
+The next statement describes the set of singular points of rank 1.
+11
+
+Theorem 3. The set of all singular points of rank 1 for the system with Hamil-
+tonian (4) and integral 𝐾 = 𝑆3 on e(3)* is given by the following two equations
+in the coordinates (𝑥, 𝑚, 𝜙, 𝑘, 𝑎, 𝑔):
+𝑚 = −(𝑎 − 𝑥2)𝑔1,
+(14)
+(𝑘𝑥−𝑔)(𝑘𝑎−𝑔𝑥)
+(𝑎 − 𝑥2)2
++ 𝑥𝑔2
+1 − (𝑎−𝑥2)𝑔1
+𝜕𝑔1
+𝜕𝑥 + 𝑔𝜕𝑔2
+𝜕𝑥 + 𝑘𝜕𝑔3
+𝜕𝑥 + 𝜕𝑉
+𝜕𝑥 = 0.
+(15)
+Proof. Calculating the matrix of the Poisson bracket in the coordinates (𝑥, 𝑚, 𝜙, 𝑘, 𝑎, 𝑔),
+one obtains
+⎛
+⎜
+⎜
+⎜
+⎜
+⎜
+⎜
+⎝
+0
+𝑎 − 𝑥2
+0
+0
+0
+0
+𝑥2 − 𝑎
+0
+0
+0
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+−1
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+⎞
+⎟
+⎟
+⎟
+⎟
+⎟
+⎟
+⎠
+.
+Therefore, in these coordinates the skew-gradients of 𝐻 and 𝐾 are
+sgrad 𝐻 =
+(︁
+(𝑎 − 𝑥2)𝜕𝐻
+𝜕𝑚, (𝑥2 − 𝑎)𝜕𝐻
+𝜕𝑥 , 𝜕𝐻
+𝜕𝑘 , 0, 0, 0
+)︁
+,
+sgrad 𝐾 = (0, 0, 1, 0, 0, 0).
+(16)
+Here we take into account that 𝜕𝐻
+𝜕𝜙 = {𝐻, 𝐾} ≡ 0.
+Thus the condition of linear dependence of sgrad 𝐻 and sgrad 𝐾 at a point
+𝑦 ∈ e(3)*
+sgrad 𝐻 = 𝜆 sgrad 𝐾
+is equivalent to the conditions
+𝜕𝐻
+𝜕𝑚 = 0,
+𝜕𝐻
+𝜕𝑥 = 0,
+𝜕𝐻
+𝜕𝑘 = 𝜆
+(17)
+at the point 𝑦. Differentiating Hamiltonian (13) with respect to 𝑚 and 𝑥, we
+see that 𝜕𝐻
+𝜕𝑚 = 0 is equivalent to (14) and 𝜕𝐻
+𝜕𝑥 = 0 is equivalent to (15) after the
+substitution of 𝑚 from (14).
+Corollary 2. On each orbit 𝑀4
+𝑎,𝑔 the set of singular points of rank 1 form a one-
+parameter family of critical circles, which is parametrized by points (𝑘, 𝑥) of curves
+defined by equation (15). For each point (𝑘, 𝑥) satisfying (15) the corresponding
+critical circle in 𝑀4
+𝑎,𝑔 is given by the formulas
+𝑆1=(𝑔−𝑘𝑥) cos 𝜙−(𝑎−𝑥2)𝑔1 sin 𝜙
+√
+𝑎 − 𝑥2
+,
+𝑆2=(𝑔−𝑘𝑥) sin 𝜙+(𝑎−𝑥2)𝑔1 cos 𝜙
+√
+𝑎 − 𝑥2
+,
+𝑆3 = 𝑘,
+𝑅1 =
+√︀
+𝑎 − 𝑥2 cos 𝜙,
+𝑅2 =
+√︀
+𝑎 − 𝑥2 sin 𝜙,
+𝑅3 = 𝑥,
+where 𝜙 is a parameter on the circle.
+Proof. As it is shown in the proof of Theorem 3, sgrad 𝐾 =
+𝜕
+𝜕𝜙 in the coordi-
+nates (𝑥, 𝑚, 𝜙, 𝑘, 𝑎, 𝑔). Therefore, each critical circle is a coordinate line of the
+coordinate 𝜙.
+Substituting (14) into expressions (12), we obtain the required
+formulas.
+12
+
+Now we can describe the bifurcation diagram. For each pair of parameters
+𝑎, 𝑔, where 𝑎 > 0, consider the function
+𝑊𝑎,𝑔(𝑘, 𝑥) = (𝑔 − 𝑘𝑥)2
+2(𝑎 − 𝑥2) + 𝑘2
+2𝛽 − 𝑔2
+1
+2 (𝑎 − 𝑥2) + 𝑔2𝑔 + 𝑔3𝑘 + 𝑉,
+(18)
+which is an analogue of a reduced potential. Recall that 𝑔1, 𝑔2, 𝑔3, 𝑉 are functions
+of 𝑎 and 𝑥.
+Theorem 4. The bifurcation diagram of the integrable Hamiltonian system with
+Hamiltonian (4) and the integral 𝐾 = 𝑆3 on orbit (2) consists of the following
+subsets on the plane R2(ℎ, 𝑘):
+1) two points 𝑍± (they can coinside if 𝑔 = 0) with coordinates
+ℎ = 𝑔2
+2𝛽𝑎 + 𝑔 𝑔2(𝑎, ±√𝑎) ± 𝑔
+√𝑎 𝑔3(𝑎, ±√𝑎) + 𝑉 (𝑎, ±√𝑎),
+𝑘 = ± 𝑔
+√𝑎,
+which are the images of two singular points of rank 0;
+2) the points (ℎ(𝑥), 𝑘(𝑥)) which are the images of singular points of rank 1 and
+are parametrized by the parameter 𝑥, where the function 𝑘(𝑥) is implicitly defined
+by the quadratic (or linear) equation 𝜕𝑊𝑎,𝑔
+𝜕𝑥 (𝑘, 𝑥) = 0, and ℎ(𝑥) = 𝑊𝑎,𝑔(𝑘(𝑥), 𝑥).
+Proof. The first statement immediately follows from Theorem 1 describing singu-
+lar points of rank 0. Similarly, the second one follows from Theorem 3 describing
+singular points of rank 1 by taking into account expression (13) for the Hamilto-
+nian 𝐻 and definition (18) of the function 𝑊𝑎,𝑔.
+Remark 3. For each fixed 𝑎, 𝑔 the equations from Theorem 4
+ℎ = 𝑊𝑎,𝑔(𝑘, 𝑥),
+𝜕𝑊𝑎,𝑔
+𝜕𝑥
+(𝑘, 𝑥) = 0
+(19)
+describing the image of the set of singular points of rank 1 belonging to the orbit
+𝑀4
+𝑎,𝑔 are exactly the equations for the envelope of the family of parabolas
+ℎ =
+(︁
+𝑥2
+2(𝑎 − 𝑥2) + 1
+2𝛽
+)︁
+𝑘2 + 𝐵𝑎,𝑔(𝑥)𝑘 + 𝐶𝑎,𝑔(𝑥)
+on the plane R2(ℎ, 𝑘) depending on the parameter 𝑥, where
+𝐵𝑎,𝑔(𝑥) = 𝑔3(𝑎, 𝑥) −
+𝑔𝑥
+𝑎 − 𝑥2 ,
+𝐶𝑎,𝑔(𝑥) =
+𝑔2
+2(𝑎 − 𝑥2) − 𝑔2
+1(𝑎, 𝑥)
+2
+(𝑎 − 𝑥2) + 𝑔2(𝑎, 𝑥)𝑔 + 𝑉 (𝑎, 𝑥)
+(20)
+(see formula (18)). In other words, the bifurcation diagram (without points 𝑍±)
+can be regarded as the envelope of this family of parabolas.
+The bifurcation diagram Σ is the union of Σ0 = {𝑍±} and Σ1 which consists of
+the images of singular points of rank 1. Let us rewrite conditions (19) describing
+Σ1 in a more explicit parametric form.
+13
+
+The relation 𝜕𝑊𝑎,𝑔
+𝜕𝑥 (𝑘, 𝑥) = 0 from Theorem 4 is exactly equation (15). In
+notation (20) it can be written as
+𝑎𝑥
+(𝑎 − 𝑥2)2 𝑘2 + 𝐵′
+𝑎,𝑔(𝑥)𝑘 + 𝐶′
+𝑎,𝑔(𝑥) = 0,
+(21)
+where
+𝐵′
+𝑎,𝑔(𝑥) = 𝜕𝑔3
+𝜕𝑥 − 𝑔(𝑎 + 𝑥2)
+(𝑎 − 𝑥2)2 ,
+𝐶′
+𝑎,𝑔(𝑥) =
+𝑔2𝑥
+(𝑎 − 𝑥2)2 + 𝑥𝑔2
+1 − (𝑎 − 𝑥2)𝑔1
+𝜕𝑔1
+𝜕𝑥 + 𝑔𝜕𝑔2
+𝜕𝑥 + 𝜕𝑉
+𝜕𝑥 .
+Equation (21) is quadratic with respect to 𝑘 for 𝑥 ̸= 0 (it is reduced to linear
+equation for 𝑥 = 0). Its discriminant equals
+𝐷𝑎,𝑔(𝑥) = (𝐵′
+𝑎,𝑔(𝑥))2 −
+4𝑎𝑥
+(𝑎−𝑥2)2 𝐶′
+𝑎,𝑔(𝑥) =
+1
+(𝑎−𝑥2)2
+(︁
+𝑔 − (𝑎+𝑥2)𝜕𝑔3
+𝜕𝑥
+)︁2
+−
+−
+4𝑎𝑥
+(𝑎 − 𝑥2)2
+(︁
+𝑥𝑔2
+1 − (𝑎 − 𝑥2)𝑔1
+𝜕𝑔1
+𝜕𝑥 + 𝑔𝜕𝑔2
+𝜕𝑥 + 𝑥
+(︁𝜕𝑔3
+𝜕𝑥
+)︁2
++ 𝜕𝑉
+𝜕𝑥
+)︁
+.
+In order to describe a parametrization of bifurcational curves consider the set
+Θ𝑎,𝑔 = {𝑥 ∈ R | 𝑥2 < 𝑎, 𝑥 ̸= 0, 𝐷𝑎,𝑔(𝑥) ≥ 0}.
+Each its (arcwise) connected component is an interval, which is either non-dege-
+nerate (i.e., has a non-zero length) or degenerate (i.e., is a point). Denote the set
+of all non-degenerate intervals by ℐ𝑎,𝑔 and denote the set of degenerate intervals
+by Θ0
+𝑎,𝑔. Clearly, Θ𝑎,𝑔 ∖ Θ0
+𝑎,𝑔 = ⋃︀
+𝐼∈ℐ𝑎,𝑔 𝐼.
+Since Θ𝑎,𝑔 is, evidently, a closed subset of (−√𝑎, 0) ∪ (0, √𝑎), intervals from
+ℐ𝑎,𝑔 contain their endpoints except for the case when an endpoint is ±√𝑎 or 0.
+Thus, the set Σ1 in the plane R2(ℎ, 𝑘) contains curves defined on intervals
+from ℐ𝑎,𝑔, “separate” points corresponding to points from Θ0
+𝑎,𝑔, and, possibly,
+something else corresponding to 𝑥 = 0. An explicite description of Σ1 is given in
+the following statement.
+Theorem 5. The set Σ1 for the integrable Hamiltonian system with Hamiltonian
+(4) and the integral 𝐾 = 𝑆3 on orbit (2) is the union of the following parametric
+curves and points on the plane R2(ℎ, 𝑘):
+1) the pairs of curves (ℎ±(𝑥), 𝑘±(𝑥)), 𝑥 ∈ 𝐼, for each 𝐼 ∈ ℐ𝑎,𝑔, where
+ℎ±(𝑥) = (𝑔−𝑘±(𝑥)𝑥)2
+2(𝑎 − 𝑥2)
++𝑘2
+±(𝑥)
+2𝛽
+− (𝑎−𝑥2)𝑔2
+1
+2
++ 𝑔2𝑔 + 𝑔3𝑘±(𝑥) + 𝑉,
+𝑘±(𝑥) = 𝑔(𝑎 + 𝑥2)
+2𝑎𝑥
+− (𝑎 − 𝑥2)2
+2𝑎𝑥
+𝜕𝑔3
+𝜕𝑥 ± (𝑎 − 𝑥2)
+2𝑎𝑥
+×
+×
+√︂(︁
+𝑔−(𝑎+𝑥2)𝜕𝑔3
+𝜕𝑥
+)︁2
+−4𝑎𝑥
+(︁
+𝑥𝑔2
+1−(𝑎−𝑥2)𝑔1
+𝜕𝑔1
+𝜕𝑥 +𝑔𝜕𝑔2
+𝜕𝑥 +𝑥
+(︁𝜕𝑔3
+𝜕𝑥
+)︁2
++𝜕𝑉
+𝜕𝑥
+)︁
+;
+(22)
+2) the points (ℎ(𝑥0), 𝑘(𝑥0)) for each 𝑥0 ∈ Θ0
+𝑎,𝑔, where
+ℎ(𝑥0) = (𝑔−𝑘(𝑥0)𝑥0)2
+2(𝑎 − 𝑥2
+0)
++ 𝑘2(𝑥0)
+2𝛽
+− (𝑎−𝑥2
+0)𝑔2
+1
+2
++ 𝑔2𝑔 + 𝑔3𝑘(𝑥0) + 𝑉,
+𝑘(𝑥0) = 𝑔(𝑎 + 𝑥2
+0)
+2𝑎𝑥0
+− (𝑎 − 𝑥2
+0)2
+2𝑎𝑥0
+𝜕𝑔3
+𝜕𝑥 (𝑎, 𝑥0),
+14
+
+and 𝑔1, 𝑔2, 𝑔3, 𝑉 in these formulas mean the values of the corresponding functions
+at the point (𝑎, 𝑥0);
+3) for the orbits 𝑀4
+𝑎,𝑔, where 𝑔 ̸= 𝑎 𝜕𝑔3
+𝜕𝑥 (𝑎, 0), the point (ℎ0, 𝑘0), where
+ℎ0 = 𝑔2
+2𝑎 + 𝑘2
+0
+2𝛽 − 𝑎𝑔2
+1(𝑎, 0)
+2
++ 𝑔2(𝑎, 0)𝑔 + 𝑔3(𝑎, 0)𝑘0 + 𝑉 (𝑎, 0),
+𝑘0 = 𝑎𝑔1(𝑎, 0) 𝜕𝑔1
+𝜕𝑥 (𝑎, 0) − 𝑔 𝜕𝑔2
+𝜕𝑥 (𝑎, 0) − 𝜕𝑉
+𝜕𝑥 (𝑎, 0)
+𝜕𝑔3
+𝜕𝑥 (𝑎, 0) − 𝑔
+𝑎
+;
+4) for the orbits 𝑀4
+𝑎,𝑔, where 𝑔 = 𝑎 𝜕𝑔3
+𝜕𝑥 (𝑎, 0) and 𝑎 satisfies the relation
+𝑎𝑔1(𝑎, 0)𝜕𝑔1
+𝜕𝑥 (𝑎, 0) − 𝑎𝜕𝑔3
+𝜕𝑥 (𝑎, 0)𝜕𝑔2
+𝜕𝑥 (𝑎, 0) − 𝜕𝑉
+𝜕𝑥 (𝑎, 0) = 0,
+the parabola
+ℎ = 𝑘2
+2𝛽 +𝑔3(𝑎, 0)𝑘+𝑎
+2
+(︁𝜕𝑔3
+𝜕𝑥 (𝑎, 0)
+)︁2
+−𝑎
+2𝑔1(𝑎, 0)+𝑎𝜕𝑔3
+𝜕𝑥 (𝑎, 0)𝑔2(𝑎, 0)+𝑉 (𝑎, 0).
+Proof. All formulas in cases 1)–4) follow from equations (19) and expression (18).
+The cases 1) and 2) correspond to solutions of quadratic equation (21) for each
+parameters 𝑥 from Θ𝑎,𝑔, but in the case 2), when 𝑥 ∈ Θ0
+𝑎,𝑔, the corresponding
+discriminant 𝐷𝑎,𝑔(𝑥) vanishes, since 𝐷𝑎,𝑔 is a continuous function on (−√𝑎, √𝑎).
+The case 3) corresponds to 𝑥 = 0 in equation (21). If 𝐵′
+𝑎,𝑔(0) = 𝜕𝑔3
+𝜕𝑥 (𝑎, 0)− 𝑔
+𝑎 ̸=
+0, then −𝐶′
+𝑎,𝑔(0)/𝐵′
+𝑎,𝑔(0) is the unique solution 𝑘0 of linear equation (21) for 𝑥 = 0,
+and we obtain the point (ℎ0, 𝑘0) in the case 3). Note that if 𝐵′
+𝑎,𝑔(0) ̸= 0, then
+the discriminant 𝐷𝑎,𝑔(𝑥) is positive on some interval (−𝜀, 𝜀) and there are two
+bifurcational curves (22) defined on (−𝜀, 0) and (0, 𝜀) which tend to the point
+(ℎ0, 𝑘0) as 𝑥 → 0 and form one smooth bifurcational curve glued from two curves
+at this point.
+The case 4) also corresponds to 𝑥 = 0, but the conditions on 𝑔 and 𝑎 in the
+case 4) are equivalent to the conditions 𝐵′
+𝑎,𝑔(0) = 𝐶′
+𝑎,𝑔(0) = 0, which imply that
+an arbitrary 𝑘 is a solution of (21) for 𝑥 = 0.
+Thus, we obtain the required
+parabola in the case 4).
+Note that for arbitrary functions 𝑔1, 𝑔2, 𝑔3, 𝑉 the behavior of bifurcational
+curves described in Theorem 5 by explicit formulas can be fairly complicated.
+They can have many cusps, intersect one another or coincide on some their arcs.
+Some general properties concerning the behavior of bifurcational curves are de-
+scribed in the following statement.
+Corollary 3. 1) If 𝐽 ⊂ Θ𝑎,𝑔 is an open interval such that 𝐷𝑎,𝑔|𝐽 > 0, then
+the bifurcational curve (ℎ±(𝑥), 𝑘±(𝑥)) defined on 𝐽 by formulas (22) is a smooth
+parametric curve which is regular for all 𝑥, where ���𝑘±
+𝑑𝑥 (𝑥) ̸= 0.
+2) Exactly two arcs of the bifurcational curves described in the items 1) and 4)
+of Theorem 5 tend to infinity such that ℎ(𝑘) ∼ 𝑘2
+2𝛽 (one arc for 𝑘 → +∞ and one
+arc for 𝑘 → −∞). For the curves defined by formulas (22) these arcs correspond
+to 𝑥 → 0.
+3) For each singular point 𝑃± of rank 0 which is of center-center type (by
+Theorem 2 there can be 0, 1, or 2 such points) there are exactly two arcs of the
+bifurcational curves described by formulas (22) which tend to the corresponding
+point 𝑍± described in Theorem 4 as 𝑥 → ±√𝑎.
+15
+
+Proof. Since ℎ = ℎ±(𝑥), 𝑘 = 𝑘±(𝑥) satisfy equations (19), we have
+𝑑ℎ±
+𝑑𝑥 (𝑥) = 𝜕𝑊𝑎,𝑔
+𝜕𝑘
+(𝑘±(𝑥), 𝑥)𝑑𝑘±
+𝑑𝑥 (𝑥).
+Therefore, the parametric curve (22) is regular iff 𝑑𝑘±
+𝑑𝑥 (𝑥) ̸= 0 and can have sin-
+gularities (for example, cusps) only at points, where 𝑑𝑘±
+𝑑𝑥 = 0.
+Items 2) and 3) follow from formulas (22) by investigating the behavior of
+the parametric curves (ℎ±(𝑥), 𝑘±(𝑥)) as 𝑥 tends to 0 or ±√𝑎. Note that 𝐷𝑎,𝑔 is
+positive in a neighborhood of the points ±√𝑎 iff 𝑞 from Corollary 1 is positive for
+𝑅3 = ±√𝑎.
+5
+Liouville tori bifurcations
+All basic definitions and facts about Liouville tori bifurcations can be found in [1].
+Theorem 6. A singular point of rank 1 (described in Theorem 3 and Corollary 2)
+is non-degenerate iff 𝜕2𝑊𝑎,𝑔(𝑘,𝑥)
+𝜕𝑥2
+̸= 0, where 𝑊𝑎,𝑔(𝑘, 𝑥) is given by (18). Moreover,
+• if 𝜕2𝑊𝑎,𝑔(𝑘,𝑥)
+𝜕𝑥2
+> 0, then the type of the point is elliptic;
+• if 𝜕2𝑊𝑎,𝑔(𝑘,𝑥)
+𝜕𝑥2
+< 0, then the type of the point is hyperbolic.
+The non-degeneracy and the type of a singular point 𝑦 of rank 1 are completely
+determined by the spectrum of linearization of the Hamiltonian vector field which
+is a (non-trivial) linear combination of sgrad 𝐻 and sgrad 𝐾 vanishing at 𝑦. Thus,
+Theorem 6 follows from the following statement.
+Lemma 4. Each point 𝑦 of rank 1 (described in Theorem 3 and Corollary 2) is
+a singular point for the vector field sgrad 𝐹𝑦, where 𝐹𝑦 = 𝐻 − 𝜆𝐾 and 𝜆 = 𝜕𝐻
+𝜕𝑘
+⃒⃒
+𝑦.
+The spectrum of the linearization 𝐴𝐹𝑦 = Lin(sgrad 𝐹𝑦) at the point 𝑦 consists of
+4 zeroes and
+𝜇± = ±𝑖
+√︂
+𝜕2𝑊𝑎,𝑔(𝑘, 𝑥)
+𝜕𝑥2
+.
+Proof. The proof is by direct calculation. The Hamiltonian vector fields sgrad 𝐻
+and sgrad 𝐾 in the coordinates (𝑥, 𝑚, 𝜙, 𝑘, 𝑎, 𝑔) from Lemma 3 are given by (16),
+and at a point 𝑦 ∈ e(3)* of rank 1 conditions (17) are fulfilled. Hence for the
+function 𝐹𝑦 = 𝐻 − 𝜆𝐾, where 𝜆 = 𝜕𝐻
+𝜕𝑘
+⃒⃒
+𝑦, we have sgrad𝑦 𝐹𝑦 = 0, and therefore
+the linearization 𝐴𝐹𝑦 of the field
+sgrad 𝐹𝑦 =
+(︁
+(𝑎 − 𝑥2)𝜕𝐻
+𝜕𝑚, −(𝑎 − 𝑥2)𝜕𝐻
+𝜕𝑥 , 𝜕𝐻
+𝜕𝑘 − 𝜆, 0, 0, 0
+)︁
+at the point 𝑦 is well-defined. Taking into account conditions (17), we get the
+following equation for the spectrum of 𝐴𝐹𝑦:
+det(𝐴𝐹𝑦 − 𝜇 Id) = 𝜇4(𝑎 − 𝑥2)2 det
+(︃
+𝜕2𝐻
+𝜕𝑚𝜕𝑥 − 𝜇
+𝜕2𝐻
+𝜕𝑚2
+− 𝜕2𝐻
+𝜕𝑥2
+− 𝜕2𝐻
+𝜕𝑥𝜕𝑚 − 𝜇
+)︃
+= 0.
+16
+
+Thus the non-zero eigenvalues of 𝐴𝐹𝑦 are
+𝜇± = ±
+√︂(︁ 𝜕2𝐻
+𝜕𝑥𝜕𝑚
+)︁2
+− 𝜕2𝐻
+𝜕𝑥2
+𝜕2𝐻
+𝜕𝑚2 .
+(23)
+For the function 𝐻 given by (13) we have
+𝜕2𝐻
+𝜕𝑚2 =
+1
+𝑎 − 𝑥2 ,
+𝜕2𝐻
+𝜕𝑥𝜕𝑚 = 𝜕𝑔1
+𝜕𝑥 (𝑎, 𝑥) +
+2𝑚𝑥
+(𝑎 − 𝑥2)2 ,
+𝜕2𝐻
+𝜕𝑥2 = (𝑔2 + 𝑎𝑘2 + 𝑚2)(𝑎 + 3𝑥2) − 2𝑔𝑘𝑥(𝑥2 + 3𝑎)
+(𝑎 − 𝑥2)3
++
++𝑚𝜕2𝑔1
+𝜕𝑥2 (𝑎, 𝑥) + 𝑔𝜕2𝑔2
+𝜕𝑥2 (𝑎, 𝑥) + 𝑘𝜕2𝑔3
+𝜕𝑥2 (𝑎, 𝑥) + 𝜕2𝑉
+𝜕𝑥2 (𝑎, 𝑥).
+(24)
+Since, by Theorem 3, at a singular point we have 𝑚 = −(𝑎−𝑥2)𝑔1(𝑎, 𝑥), equalities
+(24) can be rewritten as
+𝜕2𝐻
+𝜕𝑚2 =
+1
+𝑎 − 𝑥2 ,
+𝜕2𝐻
+𝜕𝑥𝜕𝑚 = 𝜕𝑔1
+𝜕𝑥 (𝑎, 𝑥) − 2𝑥𝑔1(𝑎, 𝑥)
+𝑎 − 𝑥2
+,
+𝜕2𝐻
+𝜕𝑥2 = 𝜕2𝑊𝑎,𝑔(𝑘, 𝑥)
+𝜕𝑥2
++ (𝑎 − 𝑥2)
+(︁𝜕𝑔1
+𝜕𝑥 (𝑎, 𝑥) − 2𝑥𝑔1(𝑎, 𝑥)
+𝑎 − 𝑥2
+)︁2
+,
+(25)
+where 𝑊𝑎,𝑔(𝑘, 𝑥) is given by (18). Substituting expressions (25) into formula (23)
+we get the desired expression for 𝜇±.
+Lemma 4 and, consequently, Theorem 6 are proved.
+Theorem 7. The only possible non-degenerate Liouville tori bifurcations for the
+isoenergy surfaces 𝑄3 of the integrable Hamiltonian system with Hamiltonian (4)
+and the integral 𝐾 = 𝑆3 on orbit (2) are the so-called 𝐴 and 𝑉𝑘 bifurcations. In
+particular, if there is only one singular circle in a fiber, then the bifurcation is
+either 𝐴 or 𝐵.
+Proof. There is only one elliptic bifurcation (of type 𝐴), thus we consider hy-
+perbolic bifurcations.
+Since all critical points of rank 1 satisfy the condition
+𝑅2
+1 + 𝑅2
+2 ̸= 0, we can work in the coordinates (𝑥, 𝑚, 𝜙, 𝑘, 𝑎, 𝑔).
+Consider the inverse image of a point (ℎ0, 𝑘0) under the momentum mapping
+𝑀4
+𝑎,𝑔 → R2(ℎ, 𝑘). Then 𝜙 is arbitrary and 𝑚 is given by
+(𝑚 + (𝑎 − 𝑥2)𝑔1(𝑎, 𝑥))2
+2(𝑎 − 𝑥2)
+= ℎ0 − 𝑊𝑎,𝑔(𝑘0, 𝑥),
+(26)
+where 𝑥 satisfies the condition ℎ0 ≥ 𝑊𝑎,𝑔(𝑘0, 𝑥).
+Thus any connected component of a singular fiber for a non-degenerate sin-
+gularity is a product of 𝑆1 and a wedge sum of 𝑘 circles as in Figure 1. More
+precisely, the set in the plane (𝑚, 𝑥) given by equation (26) is homeomorphic to
+the union of circles that are joined at the points ℎ0 = 𝑊𝑎,𝑔(𝑘0, 𝑥).
+Since the singularity is non-degenerate, this is precisely the bifurcation for the
+𝑉𝑘 atom. Theorem 7 is proved.
+17
+
+Figure 1: Atom 𝑉𝑘.
+6
+Isoenergy surfaces
+For a Hamiltonian function 𝐻 on e(3)* which is a positive definite quadratic
+form in S, the topology of isoenergy surfaces is completely determined by their
+projections on the Poisson shere (for details see [1]). By Theorem 2, the projection
+is invariant under rotation around the 𝑅3-axis. As a direct consequence we get
+the following statement.
+Theorem 8. Any isoenergy surface 𝑄3 of the integrable Hamiltonian system with
+Hamiltonian (4) and the integral 𝐾 = 𝑆3 on orbit (2) is either RP3 or a disjoint
+union of 𝑘 products 𝑆1 × 𝑆2 and not more than two spheres 𝑆3.
+Proof. If the projection of 𝑄3 on the Poisson sphere is surjective, then 𝑄3 = RP3.
+Otherwise the image of the projection is the unioun of 𝑙 rings and not more than
+two disks with centers in the poles R = (0, 0, 𝑅3).
+Each ring corresponds to
+𝑆1 × 𝑆2 and each disk to 𝑆3.
+ACKNOWLEDGMENTS
+This work was supported by the Russian Sci-
+ence Foundation, project no. 17-11-01303.
+References
+[1] A.V. Bolsinov and A.T. Fomenko, Integrable Hamiltonian Systems: Geometry,
+Topology, Classification (CRC, Boca Raton, FL, 2004).
+[2] A.V. Borisov and I.S. Mamaev, Rigid body dynamics (NIC Regular Chaotic
+Dynamics,Moscow, Izhevsk, 2001) [in Russian].
+[3] A. A. Oshemkov, “Fomenko invariants for the main integrable cases of the rigid
+body motion equations”, in Topological Classification of Integrable Systems,
+Adv. Sov. Math. 6, 67–146 (1991).
+[4] A. V. Bolsinov, A. M. Izosimov, A. Yu. Konjaev, and A. A. Oshemkov, “Algebra
+and topology of integrable systems. Research problems”, Tr. Sem. Vektor.
+Tenzor. Anal. 28, 119–191 (2012).
+[5] E.O. Kantonistova, “Topological classification of integrable Hamiltonian sys-
+tems in a potential field on surfaces of revolution”, Mathematics, 207, 358–399
+(2016).
+18
+
+11/1
+OXOX .X.[6] B.S. Kruglikov, Topological classification of Leggett systems in an integrable
+case for 3He-A, Uspekhi Mat. Nauk, 46: 179–181 (1991).
+[7] M.Yu. Ivochkin, Mat. Sb., 199:6 (2008), 85–104 Topological analysis of the
+motion of an ellipsoid on a smooth plane, Mat. Sb., 199, 871–890 (2008).
+19
+

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+page_content=' Kozlov∗  and  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Oshemkov†  Abstract  Integrable systems with a linear periodic integral for the Lie algebra e(3) are  considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' One investigates singulariries of the Liouville foliation, bifurcation  diagram of the momentum mapping, transformations of Liouville tori, topology  of isoenergy surfaces and other topological properties of such systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Keywords and phrases: Integrable Hamiltonian system, periodic integral,  bifurcation diagram, momentum mapping, Liouville tori  1  Introduction  In this paper we study some topological properties of integrable Hamiltonian  systems with an 𝑆1-symmetry given by the Euler equations for the Lie algebra  e(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Probably, the most well-known example of such a system is the classical  Lagrange top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Roughly speaking, we consider a “generalized” Lagrange top which  Hamiltonian has an arbitrary potential function and linear terms in momenta,  but possesses the same 𝑆1-symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  We are interested in local and global topological properties of the Liouville  foliation defined by the system under consideration, namely, the structure of bi-  furcation diagram and transformations of Liouville tori for critical values of the  momentum mapping, non-degeneracy of equilibria and other singular points, the  topology of isoenergy surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Note that there is a number of integrable systems with periodic linear inte-  gral which are well known in mechanics and mathematical physics, which phase  topology were studied by various authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' In particular, there are Lagrange and  Kirchhoff integrable cases in rigid body dynamics (for the description of their  topology see [1–3]), the integrable case of Leggett equations describing dynamics  of spin in the superfluid 3He (the bifurcation diagram and Fomenko invariants  for this system are described in [6]), the integrable case of the motion of heavy  ellipsoid on a smooth horizontal plane (topological invariants for this system were  found in [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  ∗No Affiliation, E-mail: ikozlov90@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='com  †Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991 Russia, E-mail:  a@oshemkov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='ru  3  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='05283v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='DG]  12 Jan 2023  Topological properties of all these systems are quite similar because of an 𝑆1-  symmetry which imposes strong restrictions on the structure of their singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Therefore, they can be studied under a uniform scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' In this paper we perform  such an investigation for an example of Hamiltonian possessing a periodic linear  integral on e(3)*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Note that the problem of topological investigation of integrable  systems with S1-action is discussed in paper [4], which contains a list of various  open problems in the theory of integrable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Apart from the systems on e(3)* considered in this paper there are other  integrable systems with 𝑆1-symmetry, which were also studied by various authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  For instance, natural mechanical systems on surfaces of revolution homeomorphic  to the sphere were studied recently in [5] (see also [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Another example is the  classical Euler case in the rigid body dynamics, where the 𝑆1-action is given not  by a linear, but by a quadratic integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' The results obtained in this paper show  in particular that there are some differences between the topological properties  of the systems under consideration and other cases with an 𝑆1-symmetry (for  example, the one investigated in [5] or the Euler case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  The article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' In Section 2 we describe the systems under  consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' We start the analysis with the study of non-deneracy and types of  singular points of rank 0 in Section 3 (Corollary 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' In Section 4 we find singular  points of rank 1 (Theorem 3) and describe the bifurcation diagrams of the system  (Theorems 4 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' In Section 5 we determine types of non-degenerate points  of rank 1 (Theorem 6) and specify the corresponding Liouville tori bifurcations  (Theorem 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Finally, in Section 6 we list all possible isoenergy surfaces for the  system (Theorem 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  2  Description of the system  Let us recall that the Lie–Poisson bracket for the Lie algebra e(3) is given by the  formulas  {𝑆𝑖, 𝑆𝑗} = 𝜀𝑖𝑗𝑘𝑆𝑘,  {𝑆𝑖, 𝑅𝑗} = 𝜀𝑖𝑗𝑘𝑅𝑘,  {𝑅𝑖, 𝑅𝑗} = 0,  (1)  where 𝑆1, 𝑆2, 𝑆3, 𝑅1, 𝑅2, 𝑅3 are linear coordinates on the dual space e(3)* for the  Lie algebra e(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' We will use the notation S = (𝑆1, 𝑆2, 𝑆3) and R = (𝑅1, 𝑅2, 𝑅3)  and also ⟨·,·⟩ and × for the scalar and vector product of 3-dimensional vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  A Hamiltonian system with Hamiltonian 𝐻 is given by the Euler equations  ˙𝑥𝑖 = {𝑥𝑖, 𝐻},  which for the Lie algebra e(3) take the form  ˙S = 𝜕𝐻  𝜕S × S + 𝜕𝐻  𝜕R × R,  ˙R = 𝜕𝐻  𝜕S × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Bracket (1) has two Casimir functions:  𝐹1 = ⟨R, R⟩,  𝐹2 = ⟨S, R⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Their regular common level surfaces  𝑀4  𝑎,𝑔 = {(S, R) | 𝐹1(S, R) = 𝑎, 𝐹2(S, R) = 𝑔, },  𝑎 > 0,  (2)  4  are the sympectic leaves of bracket (1) and are the orbits of the coadjoint repsre-  sentation for the Lie algebra e(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' We are interested in integrable Hamiltonian  systems on the orbits 𝑀4  𝑎,𝑔 for which some linear function on e(3)* is a first integral  defining an 𝑆1-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Let us describe several examples of such systems from mechanics and math-  ematical physics, which are integrable cases of the Euler equations for the Lie  algebra e(3) with Hamiltonian 𝐻 and integral 𝐾 (an explanation of physical sense  for parameters and variables of these systems can be found in [1,2,6,7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  1) The Lagrange case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' This is a symmetric top with two equal moments of  inertia which center of gravity lies on the symmetry axis:  𝐻 = 𝑆2  1  𝐴 + 𝑆2  2  𝐴 + 𝑆2  3  𝐵 − 𝑝𝑅3,  𝐾 = 𝑆3,  where 𝐴, 𝐵, 𝑝 = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  2) The Kirchhoff case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' This system describes the motion of a dynamically  symmetric rigid body in an ideal fluid:  𝐻 = 𝐴𝑆2  1 + 𝐴𝑆2  2 + 𝑎𝑆2  3 + 2(𝐵𝑆1𝑅1 + 2𝐵𝑆2𝑅2 + 𝑏𝑆3𝑅3)+  + 𝐶𝑅2  1 + 𝐶𝑅2  2 + 𝑐𝑅2  3,  𝐾 = 𝑆3,  where 𝐴, 𝑎, 𝐵, 𝑏, 𝐶, 𝑐 = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  3) The following integrable case for the Leggett system describing the dynamics  of spin in the superfluid 3He:  𝐻 = 𝑆2  1 + 𝑆2  2 + 𝑆2  3 − 𝛾𝑆3 − 𝑅2  3,  𝐾 = 𝑆3,  where 𝛾 = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  4) Integrable system describing the motion of a dynamically and geometrically  symmetric heavy ellipsoid on a smooth horizontal plane:  𝐻 = 𝑆2  1 + 𝑆2  2 + 𝐴(𝑆1𝑅1 + 𝑆2𝑅2)2  2𝑏(1 + 𝐴(𝑅2  1 + 𝑅2  2))  + 𝑆2  3  2𝐽 +  √︁  1 + 𝑐𝑅2  3 + 𝑠𝑅3,  𝐾 = 𝑆3  where 𝐴 =  𝑐𝑅2  3  1 + 𝑐𝑅2  3  ,  𝑏, 𝑐, 𝐽, 𝑠 = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  In all these examples the additional integral is the function 𝑆3 on e(3)*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Let  us explain that this is a general case if we require that the integral is linear and  periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Assertion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Let 𝐾 be a linear functions on e(3)* which Hamiltonian flow sgrad 𝐾  defined by bracket (1) is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Then there is a linear change of variables pre-  serving the bracket (1) taking the function 𝐾 to 𝑐𝑆3, where 𝑐 is some constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Let 𝐾 = 𝛼1𝑆1 + 𝛼2𝑆2 + 𝛼3𝑆3 + 𝛽1𝑅1 + 𝛽2𝑅2 + 𝛽3𝑅3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' For an arbitrary or-  thogonal matrix 𝐴 the transformation Φ𝐴 : (S, R) → (𝐴S, 𝐴R) preserves bracket  (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  If 𝛼1 = 𝛼2 = 𝛼3 = 0, then we can choose a matrix 𝐴 such that Φ𝐴  takes the function 𝐾 to 𝜆𝑅3, where 𝜆 = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' It is clear that the Hamilto-  nian flow of the function 𝜆𝑅3 is not periodic, since the trajectories of the field  sgrad 𝑅3 = (−𝑅2, 𝑅1, 0, 0, 0, 0) are straight lines in e(3)*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  If there are non-zero 𝛼𝑖, then applying an appropriate transformation Φ𝐴 we  can transform 𝐾 to a function of the form 𝑐𝑆3 + 𝛽′  1𝑅1 + 𝛽′  2𝑅2 + 𝛽′  3���3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' It is easy  to check that for any vector v the transformations Ψv : (S, R) → (S + v × R, R)  5  also preserve bracket (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' This allows one to transform the function 𝐾 to the form  𝑐𝑆3 + 𝜆𝑅3, where 𝑐 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Now consider the function 𝐾 = 𝑆3 + 𝜆𝑅3 and determine for which 𝜆 the  Hamiltonian flow of 𝐾 is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Integral trajectories for the field sgrad 𝐾 =  (−𝑆2 − 𝜆𝑅2, 𝑆1 + 𝜆𝑅1, 0, −𝑅2, 𝑅1, 0) can be explicitly written:  𝛾(𝑡) = ((𝑠1−𝜆𝑟2𝑡) cos 𝑡−(𝑠2+𝜆𝑟1𝑡) sin 𝑡, (𝑠2+𝜆𝑟1𝑡) cos 𝑡+(𝑠1−𝜆𝑟2𝑡) sin 𝑡,  𝑠3, 𝑟1 cos 𝑡 − 𝑟2 sin 𝑡, 𝑟2 cos 𝑡 + 𝑟1 sin 𝑡, 𝑟3),  where 𝑠1, 𝑠2, 𝑠3, 𝑟1, 𝑟2, 𝑟3 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  It is clear from this formula that the  trajectories are periodic only for 𝜆 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' It is well known that an action of any compact group can be linearized  at a fixed point and that for an action of the circle 𝑆1 the corresponding tangent  space can be represented as a sum of invariant two-dimensional subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Thus  among all linear functions on e(3)* the periodic integrals are distiguished by the  property that their linearization at any singular point is a unitary operator with  respect to a complex structure on the tangent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' It also follows that up to  the choice of the coordinate system and multipltication by a constant any periodic  linear integral on e(3)* is 𝑆3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Further we will consider Hamiltonian systems for the Lie algebra e(3) which  possess the first integral 𝐾 = 𝑆3 and which Hamiltonian 𝐻 is quadratic in 𝑆, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=',  𝐻 = 𝐴1𝑆2  1 + 𝐴2𝑆2  2 + 𝐴3𝑆2  3 + 𝑓1(R)𝑆1 + 𝑓2(R)𝑆2 + 𝑓3(R)𝑆3 + 𝑓4(R),  (3)  where 𝐴1, 𝐴2, 𝐴3 are arbitrary positive constants and 𝑓1, 𝑓2, 𝑓3, 𝑓4 are smooth  functions of 𝑅1, 𝑅2, 𝑅3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  First of all, let us rewrite Hamiltonian (3) in a more convient way using its  commutativity with the function 𝑆3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Assertion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Up to multiplication by a constant any Hamiltonian of the form (3)  commuting with the function 𝐾 = 𝑆3 has the form  𝐻 = 1  2  (︁  𝑆2  1 + 𝑆2  2 + 𝑆2  3  𝛽  )︁  + 𝑔1(R2, 𝑅3)(𝑆1𝑅2 − 𝑆2𝑅1)+  + 𝑔2(R2, 𝑅3)⟨S, R⟩ + 𝑔3(R2, 𝑅3)𝑆3 + 𝑉 (R2, 𝑅3),  (4)  where 𝛽 > 0 and the functions 𝑔1, 𝑔2, 𝑔3, 𝑉 depend only on R2 and 𝑅3 and are  smooth if R2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' The Hamiltonian vector field for the function 𝐾 is equal to  sgrad 𝐾 = −𝑅2  𝜕  𝜕𝑅1  + 𝑅1  𝜕  𝜕𝑅2  − 𝑆2  𝜕  𝜕𝑆1  + 𝑆1  𝜕  𝜕𝑆2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Since {𝐻, 𝐾} = (sgrad 𝐾)𝐻 = 0, we get  (sgrad 𝐾)𝐻 = 2(𝐴2 − 𝐴1)𝑆1𝑆2+  +  (︁  −𝑅2  𝜕𝑓1  𝜕𝑅1  +𝑅1  𝜕𝑓1  𝜕𝑅2  +𝑓2(R)  )︁  𝑆1 +  (︁  −𝑅2  𝜕𝑓2  𝜕𝑅1  +𝑅1  𝜕𝑓2  𝜕𝑅2  −𝑓1(R)  )︁  𝑆2+  +  (︁  −𝑅2  𝜕𝑓3  𝜕𝑅1  + 𝑅1  𝜕𝑓3  𝜕𝑅2  )︁  𝑆3 +  (︁  −𝑅2  𝜕𝑓4  𝜕𝑅1  + 𝑅1  𝜕𝑓4  𝜕𝑅2  )︁  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  6  Hence, 𝐴1 = 𝐴2 (multiplying by a constant we can make both these constants  equal to 1  2) and the four expressions in the brackets are equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  In polar coordinates (𝜌, 𝜙) on the plane (𝑅1, 𝑅2) the vector field  𝜕  𝜕𝜙 is exactly  −𝑅2  𝜕  𝜕𝑅1 + 𝑅1  𝜕  𝜕𝑅2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Therefore,  𝜕𝑓3  𝜕𝜙 = 0,  𝜕𝑓4  𝜕𝜙 = 0,  𝜕𝑓1  𝜕𝜙 = −𝑓2,  𝜕𝑓2  𝜕𝜙 = 𝑓1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  The first two of these equations imply that 𝑓3 and 𝑓4 depend only on 𝜌 and  𝑅3 or, equivalently, 𝑓3(R) = 𝑔3(R2, 𝑅3) and 𝑓4(R) = 𝑉 (R2, 𝑅3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' The latter two  equations can be cosidered as a system of ODE with parameters 𝜌 and 𝑅3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Solving  it, we obtain  𝑓1 = 𝑓11(𝜌, 𝑅3) cos 𝜙 + 𝑓12(𝜌, 𝑅3) sin 𝜙 = 𝑓11(𝜌, 𝑅3)  𝜌  𝑅1 + 𝑓12(𝜌, 𝑅3)  𝜌  𝑅2,  𝑓2 = −𝑓12(𝜌, 𝑅3) cos 𝜙+𝑓11(𝜌, 𝑅3) sin 𝜙 = −𝑓12(𝜌, 𝑅3)  𝜌  𝑅1+𝑓11(𝜌, 𝑅3)  𝜌  𝑅2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Since 𝜌 =  √︀  𝑅2  1 + 𝑅2  2 we get the desired form for the Hamiltonian 𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  3  Singularities of rank 0  It turns out that equilibria points for a Hamiltonian system on e(3)* possessing  a linear periodic integral 𝐾 are exactly the points where sgrad 𝐾 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  This  gives the following simple description for singularities of rank 0 of such integrable  Hamiltonian systems (not necessarily with Hamiltonian of the form (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' The set of singular points of rank 0 for an integrable Hamiltonian  system on e(3)* with arbitrary Hamiltonian 𝐻 possessing the integral 𝐾 = 𝑆3 is  the two-dimensional subspace  {(0, 0, 𝑆3, 0, 0, 𝑅3)}  (5)  in e(3)*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' In particular, for each orbit 𝑀4  𝑎,𝑔 there are precisely two singular points  of rank 0:  (︁  0, 0, ± 𝑔  √𝑎, 0, 0, ±√𝑎  )︁  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' The Hamiltonian vector field of a function 𝑓 on e(3)* has the form  sgrad 𝑓 =  (︁𝜕𝑓  𝜕S × S + 𝜕𝑓  𝜕R × R, 𝜕𝑓  𝜕S × R  )︁  ,  (6)  and for the function 𝐾 = 𝑆3 we have sgrad 𝐾 = (−𝑆2, 𝑆1, 0, −𝑅2, 𝑅1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' There-  fore, sgrad 𝐾 = 0 exactly at points (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Thus, points other than (5) can not be  singular points of rank 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Let us prove that sgrad 𝐻 vanishes at points (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' The functions 𝐻 and 𝐾  commute with respect to bracket (1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=', 𝑑𝑦𝐻(sgrad𝑦 𝐾) = 0 for any point 𝑦 ∈  e(3)* (the index 𝑦 in 𝑑𝑦𝑓 or sgrad𝑦 𝑓 denotes the point at which the differential  7  or, respectively, skew-gradient of the function 𝑓 is taken).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Taking the differential  of the function 𝑑𝑦𝐻(sgrad𝑦 𝐾) at any point 𝑦 = (0, 0, 𝑆3, 0, 0, 𝑅3), we get  𝐴*  𝐾(𝑑𝑦𝐻) = 0,  (7)  where 𝐴𝐾 is the linearization operator for the vector field sgrad 𝐾 at the point 𝑦,  since sgrad𝑦 𝐾 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' The matrix of the operator 𝐴𝐾 : e(3)* → e(3)* has the form  ⎛  ⎜  ⎜  ⎜  ⎜  ⎜  ⎜  ⎝  0  −1  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  −1  0  0  0  0  1  0  0  0  0  0  0  0  0  ⎞  ⎟  ⎟  ⎟  ⎟  ⎟  ⎟  ⎠  and therefore condition (7) implies that 𝜕𝐻  𝜕𝑆1 = 𝜕𝐻  𝜕𝑆2 = 𝜕𝐻  𝜕𝑅1 = 𝜕𝐻  𝜕𝑅2 = 0 at any point  𝑦 = (0, 0, 𝑆3, 0, 0, 𝑅3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Hence sgrad 𝐻 vanishes at points (5), since at a point  𝑦 = (0, 0, 𝑆3, 0, 0, 𝑅3) formula (6) becomes  sgrad𝑦 𝑓 =  (︁  𝑆3  𝜕𝑓  𝜕𝑆2  + 𝑅3  𝜕𝑓  𝜕𝑅2  , −𝑆3  𝜕𝑓  𝜕𝑆1  − 𝑅3  𝜕𝑓  𝜕𝑅1  , 0, 𝑅3  𝜕𝑓  𝜕𝑆2  , −𝑅3  𝜕𝑓  𝜕𝑆1  , 0  )︁  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Theorem 1 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Now, let us state when these zero-rank points are non-degenerate and deter-  mine their type (for more information about non-degeneracy of singular points of  a momentum mapping see [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' For an integrable Hamiltonian system on e(3)* with arbitrary Hamil-  tonian 𝐻 possessing the integral 𝐾 = 𝑆3, the singular point of rank 0  𝑃± =  (︁  0, 0, ± 𝑔  √𝑎, 0, 0, ±√𝑎  )︁  on the orbit 𝑀4  𝑎,𝑔 is non-degenerate iff 𝑞 ̸= 0, where  𝑞 = 𝑝2 + 𝑅2  3(𝐻11𝐻22 − |𝐻12|2),  (8)  𝑝 =  𝑔  2𝑅3  𝜕2𝐻  𝜕𝑆2  1  + 𝑅3  𝜕2𝐻  𝜕𝑆1𝜕𝑅1  − 𝜕𝐻  𝜕𝑆3  ,  (9)  and  𝐻11 = 𝜕2𝐻  𝜕𝑆2  1  ,  𝐻12 =  (︁  𝜕2𝐻  𝜕𝑆1𝜕𝑅1  − 1  𝑅3  𝜕𝐻  𝜕𝑆3  )︁  + 𝑖  𝜕2𝐻  𝜕𝑆2𝜕𝑅1  ,  𝐻22 = 𝜕2𝐻  𝜕𝑅2  1  + 𝑔  𝑅3  3  𝜕𝐻  𝜕𝑆3  − 1  𝑅3  𝜕𝐻  𝜕𝑅3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Also, if the point 𝑃± is non-degenerate, then its type is  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' center-center if 𝑞 > 0,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' focus-focus if 𝑞 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Theorem 2 holds for any Hamiltonian 𝐻 that commutes (and is functionally  independent) with 𝐾 = 𝑆3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' For the Hamiltonian 𝐻 quadratic in S the condition  of non-degeneracy and types of singular points of rank 0 are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  8  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' For Hamiltonian (4) the type of singular points of rank 0 is com-  pletely determined as in Theorem 2 by  𝑞 = 𝑔2  4𝑅2  3  − 𝑅2  3𝑔2  1(𝑎, 𝑅3) + 𝑔𝑅3  𝜕𝑔2  𝜕𝑅3  (𝑎, 𝑅3) − 𝑔 ����𝑔3  𝜕𝑅3  (𝑎, 𝑅3) − 𝑅3  𝜕𝑉  𝜕𝑅3  (𝑎, 𝑅3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Calculating all expressions from Theorem 2, we have  𝐻11 = 1,  𝐻12 = − 1  𝑅3  (︁ 𝑔  𝛽𝑅3  + 𝑔3(𝑎, 𝑅3)  )︁  − 𝑖𝑔1(𝑎, 𝑅3),  𝐻22 = 𝑔  𝑅3  3  (︁ 𝑔  𝛽𝑅3  + 𝑔3(𝑎, 𝑅3)  )︁  −  − 1  𝑅3  (︁  𝑔 𝜕𝑔2  𝜕𝑅3  (𝑎, 𝑅3) + 𝑔  𝑅3  𝜕𝑔3  𝜕𝑅3  (𝑎, 𝑅3) + 𝜕𝑉  𝜕𝑅3  (𝑎, 𝑅3)  )︁  ,  (10)  and  𝑝 = 𝑔  𝑅3  (︁1  2 − 1  𝛽  )︁  − 𝑔3(𝑎, 𝑅3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Substituting them into (8), one obtains the required formula for 𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  In order to prove Theorem 2 we use the following criteria of non-degeneracy  (see [1]), which can be regarded as a definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' A point 𝑃 of rank 0 for an integrable Hamiltonian system with  Hamiltonian 𝐻 and integral 𝐾 on a symplectic manifold 𝑀4 is non-degenerate iff  the following two conditions hold:  the linearizations 𝐴𝐻 and 𝐴𝐾 of the Hamiltonian vector fields sgrad 𝐻 and  sgrad 𝐾 at the point 𝑃 are linear independent,  there exists a linear combination 𝜆𝐴𝐻 + 𝜇𝐴𝐾 with four different non-zero  eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Let us study the spectrum of linearization of sgrad 𝐻 at the points of rank 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Taking functions 𝑆1, 𝑆2, 𝑅1, 𝑅2 as local coordinates in a neighbourhood of 0-rank  point 𝑃± on an orbit 𝑀4  𝑎,𝑔 we have  𝑅3 = ±  √︁  𝑎 − 𝑅2  1 − 𝑅2  2,  𝑆3 = 1  𝑅3  (𝑔 − 𝑆1𝑅1 − 𝑆2𝑅2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Denote by ̂︀𝐻(𝑆1, 𝑆2, 𝑅1, 𝑅2) the restriction of the fucntion 𝐻 onto 𝑀4  𝑎,𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' For any function 𝐻 commuting with 𝐾 = 𝑆3 the spectrum of the  linearization operator 𝐴 ̂︀  𝐻 = Lin(sgrad ̂︀𝐻) at the singular points 𝑃± of rank 0 has  the form 𝜎(𝐴 ̂︀  𝐻) = {±𝑖(𝑝 + √𝑞), ±𝑖(𝑝 − √𝑞)}, where 𝑝 and 𝑞 are given by (9) and  (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' In the coordinates 𝑆1, 𝑆2, 𝑅1, 𝑅2 the Poisson bracket on the symplectic leaf  𝑀4  𝑎,𝑔 has the form  𝒜 =  ⎛  ⎜  ⎜  ⎝  0  𝑆3  0  𝑅3  −𝑆3  0  −𝑅3  0  0  𝑅3  0  0  −𝑅3  0  0  0  ⎞  ⎟  ⎟  ⎠ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  9  It is easy to check that the linearization of sgrad 𝐾 defines a complex structure  on the tangent space:  𝐴 ̂︀  𝐾 = Lin(sgrad ̂︀𝐾) =  ⎛  ⎜  ⎜  ⎝  0  −1  0  0  1  0  0  0  0  0  0  −1  0  0  1  0  ⎞  ⎟  ⎟  ⎠ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  (11)  Since [𝐴 ̂︀  𝐻, 𝐴 ̂︀  𝐾] = 0, the operator 𝐴 ̂︀  𝐻 can be complexified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' The matrix of  the Poisson structure can also be complexified, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=', we can identify (2 × 2)-blocks  (︁  𝛼 −𝛽  𝛽  𝛼  )︁  in matrices with complex numbers 𝛼 + 𝑖𝛽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Thus, in the complex coordi-  nates 𝑆1 + 𝑖𝑆2, 𝑅1 + 𝑖𝑅2 the matrix 𝒜 of the Poisson structure has the form  𝒜 =  (︂−𝑖𝑆3  −𝑖𝑅3  −𝑖𝑅3  0  )︂  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  On a symplectic manifold we have 𝐴 ̂︀  𝐻 = 𝒜 𝑑2 ̂︀𝐻, and therefore 𝑑2 ̂︀𝐻 can also be  complexified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' By direct calculation we get  𝑑2 ̂︀𝐻 =  (︂𝐻11  𝐻12  𝐻12  𝐻22  )︂  ,  where 𝐻𝑙𝑗 are given by formulas (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' The imaginary parts of 𝐻11 and 𝐻22 vanish  because 𝐻 commutes with 𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Using the fact that if 𝜇1, 𝜇2 are eigenvalues of a matrix (𝐴 + 𝑖𝐵) for real ma-  trices 𝐴, 𝐵, then the matrix  (︀ 𝐴  𝐵  −𝐵 𝐴  )︀  has the eigenvalues 𝜇1, 𝜇2, 𝜇1, 𝜇2, we obtain  that the specturm of the (real) operator 𝐴 ̂︀  𝐻 is given by the equation  𝜇2 − 𝑖(𝑆3𝐻11 + 𝑅3𝐻12 + 𝑅3𝐻12)𝜇 + 𝑅2  3(𝐻11𝐻22 − |𝐻12|2) = 0,  which solutions give the desired spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Lemma 1 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' It is clear from (11) that for the integral 𝐾 = 𝑆3 the spectrum of the  corresponding operator 𝐴 ̂︀  𝐾 is 𝜎(𝐴 ̂︀  𝐾) = {𝑖, −𝑖, 𝑖, −𝑖}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' This doesn’t immediately  prove non-deneracy of points but shows that non-degenerate points can be only of  center-center or focus-focus type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Using Lemma 1 and Definition 1 of non-degeneracy we get  the condition of the theorem in all cases except for 𝑞 = 0 or 𝑝2 = 𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  If 𝑞 = 0, then the spectra of 𝐴 ̂︀  𝐻 and 𝐴 ̂︀  𝐾 are proportional, thus the point is  degenerate (this is precisely the moment when the image of a focus-focus point  meets an arc of the bifurcation diagram while transforming into a center-center  point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  If 𝑝2 = 𝑞, then the point is non-degenerate, and one should just take another  linear combination with different eigenvalues (such a linear combination exists  since the spectra of 𝐴 ̂︀  𝐻 and 𝐴 ̂︀  𝐾 are non-proportional).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  10  4  Bifurcation diagrams  In order to construct the bifurcation diagram let us describe all critical points of  the momentum mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' The singular points of rank 0 are found in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Thus, it remains to describe only singular points of rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' The next two lemmas  show that we can use some convenient coordinates for investigating them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' For a Hamiltonian system with Hamiltonian 𝐻 of the form (4) and  integral 𝐾 = 𝑆3, the subspace {(S, R) | 𝑅1 = 𝑅2 = 0} in e(3)* does not contain  points of rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Since we know all singular points of rank 0 (they are points with 𝑅1 = 𝑅2 =  𝑆1 = 𝑆2 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' see Theorem 1), it suffices to prove that if 𝑦 = (𝑆1, 𝑆2, 𝑆3, 0, 0, 𝑅3) ∈  e(3)* is a singular point, then its coordinates 𝑆1 and 𝑆2 vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Suppose that  this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Then sgrad𝑦 𝐾 = (−𝑆2, 𝑆1, 0, 0, 0, 0) ̸= 0 and, therefore,  sgrad𝑦 𝐻 = 𝜆 sgrad𝑦 𝐾 for a certain 𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Hence, by formula (6) (taking into account  that 𝑅3 ̸= 0), we have 𝜕𝐻  𝜕𝑆1 = 𝜕𝐻  𝜕𝑆2 = 0 at the point 𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' But for a Hamiltonian of  the form (4) this is possible only if 𝑆1 = 𝑆2 = 0 for the point 𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Now, since we can assume that 𝑅2  1 + 𝑅2  2 ̸= 0, we choose new coordinates on  the remaining set of points 𝑈 = R6(S, R) ∖ {𝑅1 = 𝑅2 = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Note that the set 𝑈  is homeomorphic to R5 × 𝑆1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Formulas  𝑆1 = (𝑔 − 𝑘𝑥) cos 𝜙 + 𝑚 sin 𝜙  √  𝑎 − 𝑥2  ,  𝑆2 = (𝑔 − 𝑘𝑥) sin 𝜙 − 𝑚 cos 𝜙  √  𝑎 − 𝑥2  ,  𝑆3 = 𝑘,  𝑅1 =  √︀  𝑎 − 𝑥2 cos 𝜙,  𝑅2 =  √︀  𝑎 − 𝑥2 sin 𝜙,  𝑅3 = 𝑥  (12)  define regular coordinates (𝑥, 𝑚, 𝜙, 𝑘, 𝑎, 𝑔) on the set 𝑈, where 𝑥2 < 𝑎 and 𝜙 is an  angular coordinate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=', is defined modulo 2𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  The inverse change of variables on the set 𝑈, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=', the expression of (𝑥, 𝑚, 𝜙, 𝑘, 𝑎, 𝑔)  through (S, R) is as follows:  𝑥 = 𝑅3,  𝑚 = 𝑀(S, R) = 𝑆1𝑅2 − 𝑆2𝑅1,  𝜙 = arg(𝑅1 + 𝑖𝑅2),  𝑘 = 𝑆3,  𝑎 = 𝐹1(S, R) = ⟨R, R⟩,  𝑔 = 𝐹2(S, R) = ⟨S, R⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' By direct calculation, it is easy to check that given formulas define a bi-  jection and that the Jacobian does not vanish on 𝑈:  det  𝜕(𝑥, 𝑚, 𝜙, 𝑘, 𝑎, 𝑔)  𝜕(𝑆1, 𝑆2, 𝑆3, 𝑅1, 𝑅2, 𝑅3) = 2(𝑅2  1 + 𝑅2  2) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Substituting expressions (12) into (4), we obtain that the Hamiltonian in the  coordinates (𝑥, 𝑚, 𝜙, 𝑘, 𝑎, 𝑔) on the set 𝑈 has the form  𝐻 = (𝑔−𝑘𝑥)2+𝑚2  2(𝑎 − 𝑥2)  + 𝑘2  2𝛽 + 𝑔1(𝑎, 𝑥)𝑚 + 𝑔2(𝑎, 𝑥)𝑔 + 𝑔3(𝑎, 𝑥)𝑘 + 𝑉 (𝑎, 𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  (13)  Futher we will often write 𝑔1, 𝑔2, 𝑔3, 𝑉 without arguments assuming that they  are functions of 𝑎 and 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  The next statement describes the set of singular points of rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  11  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' The set of all singular points of rank 1 for the system with Hamil-  tonian (4) and integral 𝐾 = 𝑆3 on e(3)* is given by the following two equations  in the coordinates (𝑥, 𝑚, 𝜙, 𝑘, 𝑎, 𝑔):  𝑚 = −(𝑎 − 𝑥2)𝑔1,  (14)  (𝑘𝑥−𝑔)(𝑘𝑎−𝑔𝑥)  (𝑎 − 𝑥2)2  + 𝑥𝑔2  1 − (𝑎−𝑥2)𝑔1  𝜕𝑔1  𝜕𝑥 + 𝑔𝜕𝑔2  𝜕𝑥 + 𝑘𝜕𝑔3  𝜕𝑥 + 𝜕𝑉  𝜕𝑥 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  (15)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Calculating the matrix of the Poisson bracket in the coordinates (𝑥, 𝑚, 𝜙, 𝑘, 𝑎, 𝑔),  one obtains  ⎛  ⎜  ⎜  ⎜  ⎜  ⎜  ⎜  ⎝  0  𝑎 − 𝑥2  0  0  0  0  𝑥2 − 𝑎  0  0  0  0  0  0  0  0  1  0  0  0  0  −1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  ⎞  ⎟  ⎟  ⎟  ⎟  ⎟  ⎟  ⎠  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Therefore, in these coordinates the skew-gradients of 𝐻 and 𝐾 are  sgrad 𝐻 =  (︁  (𝑎 − 𝑥2)𝜕𝐻  𝜕𝑚, (𝑥2 − 𝑎)𝜕𝐻  𝜕𝑥 , 𝜕𝐻  𝜕𝑘 , 0, 0, 0  )︁  ,  sgrad 𝐾 = (0, 0, 1, 0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  (16)  Here we take into account that 𝜕𝐻  𝜕𝜙 = {𝐻, 𝐾} ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Thus the condition of linear dependence of sgrad 𝐻 and sgrad 𝐾 at a point  𝑦 ∈ e(3)*  sgrad 𝐻 = 𝜆 sgrad 𝐾  is equivalent to the conditions  𝜕𝐻  𝜕𝑚 = 0,  𝜕𝐻  𝜕𝑥 = 0,  𝜕𝐻  𝜕𝑘 = 𝜆  (17)  at the point 𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Differentiating Hamiltonian (13) with respect to 𝑚 and 𝑥, we  see that 𝜕𝐻  𝜕𝑚 = 0 is equivalent to (14) and 𝜕𝐻  𝜕𝑥 = 0 is equivalent to (15) after the  substitution of 𝑚 from (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' On each orbit 𝑀4  𝑎,𝑔 the set of singular points of rank 1 form a one-  parameter family of critical circles, which is parametrized by points (𝑘, 𝑥) of curves  defined by equation (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' For each point (𝑘, 𝑥) satisfying (15) the corresponding  critical circle in 𝑀4  𝑎,𝑔 is given by the formulas  𝑆1=(𝑔−𝑘𝑥) cos 𝜙−(𝑎−𝑥2)𝑔1 sin 𝜙  √  𝑎 − 𝑥2  ,  𝑆2=(𝑔−𝑘𝑥) sin 𝜙+(𝑎−𝑥2)𝑔1 cos 𝜙  √  𝑎 − 𝑥2  ,  𝑆3 = 𝑘,  𝑅1 =  √︀  𝑎 − 𝑥2 cos 𝜙,  𝑅2 =  √︀  𝑎 − 𝑥2 sin 𝜙,  𝑅3 = 𝑥,  where 𝜙 is a parameter on the circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' As it is shown in the proof of Theorem 3, sgrad 𝐾 =  𝜕  𝜕𝜙 in the coordi-  nates (𝑥, 𝑚, 𝜙, 𝑘, 𝑎, 𝑔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Therefore, each critical circle is a coordinate line of the  coordinate 𝜙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Substituting (14) into expressions (12), we obtain the required  formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  12  Now we can describe the bifurcation diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' For each pair of parameters  𝑎, 𝑔, where 𝑎 > 0, consider the function  𝑊𝑎,𝑔(𝑘, 𝑥) = (𝑔 − 𝑘𝑥)2  2(𝑎 − 𝑥2) + 𝑘2  2𝛽 − 𝑔2  1  2 (𝑎 − 𝑥2) + 𝑔2𝑔 + 𝑔3𝑘 + 𝑉,  (18)  which is an analogue of a reduced potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Recall that 𝑔1, 𝑔2, 𝑔3, 𝑉 are functions  of 𝑎 and 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' The bifurcation diagram of the integrable Hamiltonian system with  Hamiltonian (4) and the integral 𝐾 = 𝑆3 on orbit (2) consists of the following  subsets on the plane R2(ℎ, 𝑘):  1) two points 𝑍± (they can coinside if 𝑔 = 0) with coordinates  ℎ = 𝑔2  2𝛽𝑎 + 𝑔 𝑔2(𝑎, ±√𝑎) ± 𝑔  √𝑎 𝑔3(𝑎, ±√𝑎) + 𝑉 (𝑎, ±√𝑎),  𝑘 = ± 𝑔  √𝑎,  which are the images of two singular points of rank 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  2) the points (ℎ(𝑥), 𝑘(𝑥)) which are the images of singular points of rank 1 and  are parametrized by the parameter 𝑥, where the function 𝑘(𝑥) is implicitly defined  by the quadratic (or linear) equation 𝜕𝑊𝑎,𝑔  𝜕𝑥 (𝑘, 𝑥) = 0, and ℎ(𝑥) = 𝑊𝑎,𝑔(𝑘(𝑥), 𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' The first statement immediately follows from Theorem 1 describing singu-  lar points of rank 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Similarly, the second one follows from Theorem 3 describing  singular points of rank 1 by taking into account expression (13) for the Hamilto-  nian 𝐻 and definition (18) of the function 𝑊𝑎,𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' For each fixed 𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' 𝑔 the equations from Theorem 4  ℎ = 𝑊𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='𝑔(𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' 𝑥),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  𝜕𝑊𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='𝑔  𝜕𝑥  (𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' 𝑥) = 0  (19)  describing the image of the set of singular points of rank 1 belonging to the orbit  𝑀4  𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='𝑔 are exactly the equations for the envelope of the family of parabolas  ℎ =  (︁  𝑥2  2(𝑎 − 𝑥2) + 1  2𝛽  )︁  𝑘2 + 𝐵𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='𝑔(𝑥)𝑘 + 𝐶𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='𝑔(𝑥)  on the plane R2(ℎ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' 𝑘) depending on the parameter 𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' where  𝐵𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='𝑔(𝑥) = 𝑔3(𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' 𝑥) −  𝑔𝑥  𝑎 − 𝑥2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  𝐶𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='𝑔(𝑥) =  𝑔2  2(𝑎 − 𝑥2) − 𝑔2  1(𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' 𝑥)  2  (𝑎 − 𝑥2) + 𝑔2(𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' 𝑥)𝑔 + 𝑉 (𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' 𝑥)  (20)  (see formula (18)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' In other words, the bifurcation diagram (without points 𝑍±)  can be regarded as the envelope of this family of parabolas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  The bifurcation diagram Σ is the union of Σ0 = {𝑍±} and Σ1 which consists of  the images of singular points of rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Let us rewrite conditions (19) describing  Σ1 in a more explicit parametric form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  13  The relation 𝜕𝑊𝑎,𝑔  𝜕𝑥 (𝑘, 𝑥) = 0 from Theorem 4 is exactly equation (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' In  notation (20) it can be written as  𝑎𝑥  (𝑎 − 𝑥2)2 𝑘2 + 𝐵′  𝑎,𝑔(𝑥)𝑘 + 𝐶′  𝑎,𝑔(𝑥) = 0,  (21)  where  𝐵′  𝑎,𝑔(𝑥) = 𝜕𝑔3  𝜕𝑥 − 𝑔(𝑎 + 𝑥2)  (𝑎 − 𝑥2)2 ,  𝐶′  𝑎,𝑔(𝑥) =  𝑔2𝑥  (𝑎 − 𝑥2)2 + 𝑥𝑔2  1 − (𝑎 − 𝑥2)𝑔1  𝜕𝑔1  𝜕𝑥 + 𝑔𝜕𝑔2  𝜕𝑥 + 𝜕𝑉  𝜕𝑥 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Equation (21) is quadratic with respect to 𝑘 for 𝑥 ̸= 0 (it is reduced to linear  equation for 𝑥 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Its discriminant equals  𝐷𝑎,𝑔(𝑥) = (𝐵′  𝑎,𝑔(𝑥))2 −  4𝑎𝑥  (𝑎−𝑥2)2 𝐶′  𝑎,𝑔(𝑥) =  1  (𝑎−𝑥2)2  (︁  𝑔 − (𝑎+𝑥2)𝜕𝑔3  𝜕𝑥  )︁2  −  −  4𝑎𝑥  (𝑎 − 𝑥2)2  (︁  𝑥𝑔2  1 − (𝑎 − 𝑥2)𝑔1  𝜕𝑔1  𝜕𝑥 + 𝑔𝜕𝑔2  𝜕𝑥 + 𝑥  (︁𝜕𝑔3  𝜕𝑥  )︁2  + 𝜕𝑉  𝜕𝑥  )︁  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  In order to describe a parametrization of bifurcational curves consider the set  Θ𝑎,𝑔 = {𝑥 ∈ R | 𝑥2 < 𝑎, 𝑥 ̸= 0, 𝐷𝑎,𝑔(𝑥) ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Each its (arcwise) connected component is an interval, which is either non-dege-  nerate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=', has a non-zero length) or degenerate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=', is a point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Denote the set  of all non-degenerate intervals by ℐ𝑎,𝑔 and denote the set of degenerate intervals  by Θ0  𝑎,𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Clearly, Θ𝑎,𝑔 ∖ Θ0  𝑎,𝑔 = ⋃︀  𝐼∈ℐ𝑎,𝑔 𝐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Since Θ𝑎,𝑔 is, evidently, a closed subset of (−√𝑎, 0) ∪ (0, √𝑎), intervals from  ℐ𝑎,𝑔 contain their endpoints except for the case when an endpoint is ±√𝑎 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Thus, the set Σ1 in the plane R2(ℎ, 𝑘) contains curves defined on intervals  from ℐ𝑎,𝑔, “separate” points corresponding to points from Θ0  𝑎,𝑔, and, possibly,  something else corresponding to 𝑥 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' An explicite description of Σ1 is given in  the following statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' The set Σ1 for the integrable Hamiltonian system with Hamiltonian  (4) and the integral 𝐾 = 𝑆3 on orbit (2) is the union of the following parametric  curves and points on the plane R2(ℎ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' 𝑘):  1) the pairs of curves (ℎ±(𝑥),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' 𝑘±(𝑥)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' 𝑥 ∈ 𝐼,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' for each 𝐼 ∈ ℐ𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='𝑔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' where  ℎ±(𝑥) = (𝑔−𝑘±(𝑥)𝑥)2  2(𝑎 − 𝑥2)  +𝑘2  ±(𝑥)  2𝛽  − (𝑎−𝑥2)𝑔2  1  2  + 𝑔2𝑔 + 𝑔3𝑘±(𝑥) + 𝑉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  𝑘±(𝑥) = 𝑔(𝑎 + 𝑥2)  2𝑎𝑥  − (𝑎 − 𝑥2)2  2𝑎𝑥  𝜕𝑔3  𝜕𝑥 ± (𝑎 − 𝑥2)  2𝑎𝑥  ×  ×  √︂(︁  𝑔−(𝑎+𝑥2)𝜕𝑔3  𝜕𝑥  )︁2  −4𝑎𝑥  (︁  𝑥𝑔2  1−(𝑎−𝑥2)𝑔1  𝜕𝑔1  𝜕𝑥 +𝑔𝜕𝑔2  𝜕𝑥 +𝑥  (︁𝜕𝑔3  𝜕𝑥  )︁2  +𝜕𝑉  𝜕𝑥  )︁  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  (22)  2) the points (ℎ(𝑥0), 𝑘(𝑥0)) for each 𝑥0 ∈ Θ0  𝑎,𝑔, where  ℎ(𝑥0) = (𝑔−𝑘(𝑥0)𝑥0)2  2(𝑎 − 𝑥2  0)  + 𝑘2(𝑥0)  2𝛽  − (𝑎−𝑥2  0)𝑔2  1  2  + 𝑔2𝑔 + 𝑔3𝑘(𝑥0) + 𝑉,  𝑘(𝑥0) = 𝑔(𝑎 + 𝑥2  0)  2𝑎𝑥0  − (𝑎 − 𝑥2  0)2  2𝑎𝑥0  𝜕𝑔3  𝜕𝑥 (𝑎, 𝑥0),  14  and 𝑔1, 𝑔2, 𝑔3, 𝑉 in these formulas mean the values of the corresponding functions  at the point (𝑎, 𝑥0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  3) for the orbits 𝑀4  𝑎,𝑔, where 𝑔 ̸= 𝑎 𝜕𝑔3  𝜕𝑥 (𝑎, 0), the point (ℎ0, 𝑘0), where  ℎ0 = 𝑔2  2𝑎 + 𝑘2  0  2𝛽 − 𝑎𝑔2  1(𝑎, 0)  2  + 𝑔2(𝑎, 0)𝑔 + 𝑔3(𝑎, 0)𝑘0 + 𝑉 (𝑎, 0),  𝑘0 = 𝑎𝑔1(𝑎, 0) 𝜕𝑔1  𝜕𝑥 (𝑎, 0) − 𝑔 𝜕𝑔2  𝜕𝑥 (𝑎, 0) − 𝜕𝑉  𝜕𝑥 (𝑎, 0)  𝜕𝑔3  𝜕𝑥 (𝑎, 0) − 𝑔  𝑎  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  4) for the orbits 𝑀4  𝑎,𝑔, where 𝑔 = 𝑎 𝜕𝑔3  𝜕𝑥 (𝑎, 0) and 𝑎 satisfies the relation  𝑎𝑔1(𝑎, 0)𝜕𝑔1  𝜕𝑥 (𝑎, 0) − 𝑎𝜕𝑔3  𝜕𝑥 (𝑎, 0)𝜕𝑔2  𝜕𝑥 (𝑎, 0) − 𝜕𝑉  𝜕𝑥 (𝑎, 0) = 0,  the parabola  ℎ = 𝑘2  2𝛽 +𝑔3(𝑎, 0)𝑘+𝑎  2  (︁𝜕𝑔3  𝜕𝑥 (𝑎, 0)  )︁2  −𝑎  2𝑔1(𝑎, 0)+𝑎𝜕𝑔3  𝜕𝑥 (𝑎, 0)𝑔2(𝑎, 0)+𝑉 (𝑎, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' All formulas in cases 1)–4) follow from equations (19) and expression (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  The cases 1) and 2) correspond to solutions of quadratic equation (21) for each  parameters 𝑥 from Θ𝑎,𝑔, but in the case 2), when 𝑥 ∈ Θ0  𝑎,𝑔, the corresponding  discriminant 𝐷𝑎,𝑔(𝑥) vanishes, since 𝐷𝑎,𝑔 is a continuous function on (−√𝑎, √𝑎).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  The case 3) corresponds to 𝑥 = 0 in equation (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' If 𝐵′  𝑎,𝑔(0) = 𝜕𝑔3  𝜕𝑥 (𝑎, 0)− 𝑔  𝑎 ̸=  0, then −𝐶′  𝑎,𝑔(0)/𝐵′  𝑎,𝑔(0) is the unique solution 𝑘0 of linear equation (21) for 𝑥 = 0,  and we obtain the point (ℎ0, 𝑘0) in the case 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Note that if 𝐵′  𝑎,𝑔(0) ̸= 0, then  the discriminant 𝐷𝑎,𝑔(𝑥) is positive on some interval (−𝜀, 𝜀) and there are two  bifurcational curves (22) defined on (−𝜀, 0) and (0, 𝜀) which tend to the point  (ℎ0, 𝑘0) as 𝑥 → 0 and form one smooth bifurcational curve glued from two curves  at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  The case 4) also corresponds to 𝑥 = 0, but the conditions on 𝑔 and 𝑎 in the  case 4) are equivalent to the conditions 𝐵′  𝑎,𝑔(0) = 𝐶′  𝑎,𝑔(0) = 0, which imply that  an arbitrary 𝑘 is a solution of (21) for 𝑥 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Thus, we obtain the required  parabola in the case 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Note that for arbitrary functions 𝑔1, 𝑔2, 𝑔3, 𝑉 the behavior of bifurcational  curves described in Theorem 5 by explicit formulas can be fairly complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  They can have many cusps, intersect one another or coincide on some their arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Some general properties concerning the behavior of bifurcational curves are de-  scribed in the following statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' 1) If 𝐽 ⊂ Θ𝑎,𝑔 is an open interval such that 𝐷𝑎,𝑔|𝐽 > 0, then  the bifurcational curve (ℎ±(𝑥), 𝑘±(𝑥)) defined on 𝐽 by formulas (22) is a smooth  parametric curve which is regular for all 𝑥, where 𝑑𝑘±  𝑑𝑥 (𝑥) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  2) Exactly two arcs of the bifurcational curves described in the items 1) and 4)  of Theorem 5 tend to infinity such that ℎ(𝑘) ∼ 𝑘2  2𝛽 (one arc for 𝑘 → +∞ and one  arc for 𝑘 → −∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' For the curves defined by formulas (22) these arcs correspond  to 𝑥 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  3) For each singular point 𝑃± of rank 0 which is of center-center type (by  Theorem 2 there can be 0, 1, or 2 such points) there are exactly two arcs of the  bifurcational curves described by formulas (22) which tend to the corresponding  point 𝑍± described in Theorem 4 as 𝑥 → ±√𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  15  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Since ℎ = ℎ±(𝑥), 𝑘 = 𝑘±(𝑥) satisfy equations (19), we have  𝑑ℎ±  𝑑𝑥 (𝑥) = 𝜕𝑊𝑎,𝑔  𝜕𝑘  (𝑘±(𝑥), 𝑥)𝑑𝑘±  𝑑𝑥 (𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Therefore, the parametric curve (22) is regular iff 𝑑𝑘±  𝑑𝑥 (𝑥) ̸= 0 and can have sin-  gularities (for example, cusps) only at points, where 𝑑𝑘±  𝑑𝑥 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Items 2) and 3) follow from formulas (22) by investigating the behavior of  the parametric curves (ℎ±(𝑥), 𝑘±(𝑥)) as 𝑥 tends to 0 or ±√𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Note that 𝐷𝑎,𝑔 is  positive in a neighborhood of the points ±√𝑎 iff 𝑞 from Corollary 1 is positive for  𝑅3 = ±√𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  5  Liouville tori bifurcations  All basic definitions and facts about Liouville tori bifurcations can be found in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' A singular point of rank 1 (described in Theorem 3 and Corollary 2)  is non-degenerate iff 𝜕2𝑊𝑎,𝑔(𝑘,𝑥)  𝜕𝑥2  ̸= 0, where 𝑊𝑎,𝑔(𝑘, 𝑥) is given by (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Moreover,  if 𝜕2𝑊𝑎,𝑔(𝑘,𝑥)  𝜕𝑥2  > 0, then the type of the point is elliptic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  if 𝜕2𝑊𝑎,𝑔(𝑘,𝑥)  𝜕𝑥2  < 0, then the type of the point is hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  The non-degeneracy and the type of a singular point 𝑦 of rank 1 are completely  determined by the spectrum of linearization of the Hamiltonian vector field which  is a (non-trivial) linear combination of sgrad 𝐻 and sgrad 𝐾 vanishing at 𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Thus,  Theorem 6 follows from the following statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Each point 𝑦 of rank 1 (described in Theorem 3 and Corollary 2) is  a singular point for the vector field sgrad 𝐹𝑦, where 𝐹𝑦 = 𝐻 − 𝜆𝐾 and 𝜆 = 𝜕𝐻  𝜕𝑘  ⃒⃒  𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  The spectrum of the linearization 𝐴𝐹𝑦 = Lin(sgrad 𝐹𝑦) at the point 𝑦 consists of  4 zeroes and  𝜇± = ±𝑖  √︂  𝜕2𝑊𝑎,𝑔(𝑘, 𝑥)  𝜕𝑥2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' The proof is by direct calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' The Hamiltonian vector fields sgrad 𝐻  and sgrad 𝐾 in the coordinates (𝑥, 𝑚, 𝜙, 𝑘, 𝑎, 𝑔) from Lemma 3 are given by (16),  and at a point 𝑦 ∈ e(3)* of rank 1 conditions (17) are fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Hence for the  function 𝐹𝑦 = 𝐻 − 𝜆𝐾, where 𝜆 = 𝜕𝐻  𝜕𝑘  ⃒⃒  𝑦, we have sgrad𝑦 𝐹𝑦 = 0, and therefore  the linearization 𝐴𝐹𝑦 of the field  sgrad 𝐹𝑦 =  (︁  (𝑎 − 𝑥2)𝜕𝐻  𝜕𝑚, −(𝑎 − 𝑥2)𝜕𝐻  𝜕𝑥 , 𝜕𝐻  𝜕𝑘 − 𝜆, 0, 0, 0  )︁  at the point 𝑦 is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Taking into account conditions (17), we get the  following equation for the spectrum of 𝐴𝐹𝑦:  det(𝐴𝐹𝑦 − 𝜇 Id) = 𝜇4(𝑎 − 𝑥2)2 det  (︃  𝜕2𝐻  𝜕𝑚𝜕𝑥 − 𝜇  𝜕2𝐻  𝜕𝑚2  − 𝜕2𝐻  𝜕𝑥2  − 𝜕2𝐻  𝜕𝑥𝜕𝑚 − 𝜇  )︃  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  16  Thus the non-zero eigenvalues of 𝐴𝐹𝑦 are  𝜇± = ±  √︂(︁ 𝜕2𝐻  𝜕𝑥𝜕𝑚  )︁2  − 𝜕2𝐻  𝜕𝑥2  𝜕2𝐻  𝜕𝑚2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  (23)  For the function 𝐻 given by (13) we have  𝜕2𝐻  𝜕𝑚2 =  1  𝑎 − 𝑥2 ,  𝜕2𝐻  𝜕𝑥𝜕𝑚 = 𝜕𝑔1  𝜕𝑥 (𝑎, 𝑥) +  2𝑚𝑥  (𝑎 − 𝑥2)2 ,  𝜕2𝐻  𝜕𝑥2 = (𝑔2 + 𝑎𝑘2 + 𝑚2)(𝑎 + 3𝑥2) − 2𝑔𝑘𝑥(𝑥2 + 3𝑎)  (𝑎 − 𝑥2)3  +  +𝑚𝜕2𝑔1  𝜕𝑥2 (𝑎, 𝑥) + 𝑔𝜕2𝑔2  ���𝑥2 (𝑎, 𝑥) + 𝑘𝜕2𝑔3  𝜕𝑥2 (𝑎, 𝑥) + 𝜕2𝑉  𝜕𝑥2 (𝑎, 𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  (24)  Since, by Theorem 3, at a singular point we have 𝑚 = −(𝑎−𝑥2)𝑔1(𝑎, 𝑥), equalities  (24) can be rewritten as  𝜕2𝐻  𝜕𝑚2 =  1  𝑎 − 𝑥2 ,  𝜕2𝐻  𝜕𝑥𝜕𝑚 = 𝜕𝑔1  𝜕𝑥 (𝑎, 𝑥) − 2𝑥𝑔1(𝑎, 𝑥)  𝑎 − 𝑥2  ,  𝜕2𝐻  𝜕𝑥2 = 𝜕2𝑊𝑎,𝑔(𝑘, 𝑥)  𝜕𝑥2  + (𝑎 − 𝑥2)  (︁𝜕𝑔1  𝜕𝑥 (𝑎, 𝑥) − 2𝑥𝑔1(𝑎, 𝑥)  𝑎 − 𝑥2  )︁2  ,  (25)  where 𝑊𝑎,𝑔(𝑘, 𝑥) is given by (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Substituting expressions (25) into formula (23)  we get the desired expression for 𝜇±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Lemma 4 and, consequently, Theorem 6 are proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' The only possible non-degenerate Liouville tori bifurcations for the  isoenergy surfaces 𝑄3 of the integrable Hamiltonian system with Hamiltonian (4)  and the integral 𝐾 = 𝑆3 on orbit (2) are the so-called 𝐴 and 𝑉𝑘 bifurcations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' In  particular, if there is only one singular circle in a fiber, then the bifurcation is  either 𝐴 or 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' There is only one elliptic bifurcation (of type 𝐴), thus we consider hy-  perbolic bifurcations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Since all critical points of rank 1 satisfy the condition  𝑅2  1 + 𝑅2  2 ̸= 0, we can work in the coordinates (𝑥, 𝑚, 𝜙, 𝑘, 𝑎, 𝑔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Consider the inverse image of a point (ℎ0, 𝑘0) under the momentum mapping  𝑀4  𝑎,𝑔 → R2(ℎ, 𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Then 𝜙 is arbitrary and 𝑚 is given by  (𝑚 + (𝑎 − 𝑥2)𝑔1(𝑎, 𝑥))2  2(𝑎 − 𝑥2)  = ℎ0 − 𝑊𝑎,𝑔(𝑘0, 𝑥),  (26)  where 𝑥 satisfies the condition ℎ0 ≥ 𝑊𝑎,𝑔(𝑘0, 𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Thus any connected component of a singular fiber for a non-degenerate sin-  gularity is a product of 𝑆1 and a wedge sum of 𝑘 circles as in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' More  precisely, the set in the plane (𝑚, 𝑥) given by equation (26) is homeomorphic to  the union of circles that are joined at the points ℎ0 = 𝑊𝑎,𝑔(𝑘0, 𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Since the singularity is non-degenerate, this is precisely the bifurcation for the  𝑉𝑘 atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Theorem 7 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  17  Figure 1: Atom 𝑉𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  6  Isoenergy surfaces  For a Hamiltonian function 𝐻 on e(3)* which is a positive definite quadratic  form in S, the topology of isoenergy surfaces is completely determined by their  projections on the Poisson shere (for details see [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' By Theorem 2, the projection  is invariant under rotation around the 𝑅3-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' As a direct consequence we get  the following statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' Any isoenergy surface 𝑄3 of the integrable Hamiltonian system with  Hamiltonian (4) and the integral 𝐾 = 𝑆3 on orbit (2) is either RP3 or a disjoint  union of 𝑘 products 𝑆1 × 𝑆2 and not more than two spheres 𝑆3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' If the projection of 𝑄3 on the Poisson sphere is surjective, then 𝑄3 = RP3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Otherwise the image of the projection is the unioun of 𝑙 rings and not more than  two disks with centers in the poles R = (0, 0, 𝑅3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  Each ring corresponds to  𝑆1 × 𝑆2 and each disk to 𝑆3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This work was supported by the Russian Sci-  ence Foundation, project no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
+page_content=' 17-11-01303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'}
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+Index 3 biembeddings of the complete graphs
+Juvenal F. Barajas and Timothy Sun
+Department of Computer Science
+San Francisco State University
+Abstract
+We show that the complete graphs on 24s + 21 vertices have decompositions into
+two edge-disjoint subgraphs, each of which triangulates an orientable surface.
+The
+special case where the two surfaces are homeomorphic solves a generalized Earth-
+Moon problem for that surface. Unlike previous constructions, these pairs of triangular
+embeddings are derived from index 3 current graphs.
+1
+Introduction
+There are many graph parameters that generalize the notion of planarity. Perhaps the most
+well-known of such parameters is the genus of the graph, which is the smallest value g such
+that the graph has an embedding in Sg, the orientable surface of genus g. A less-studied
+parameter is the thickness of a graph, which is the size of the smallest partition of the edges
+into planar subgraphs. A graph is said to be biembeddable in surfaces S and S′ if it can
+be decomposed into two edge-disjoint subgraphs, one of which embeds in S and the other
+embeds in S′. When S is homeomorphic to S′, we simply say that the graph is biembeddable
+in S. We consider a variant of both genus and thickness, the bigenus of a graph β(G), which
+is defined to be the smallest value g such that the graph G is biembeddable in Sg.
+The Earth-Moon problem is a longstanding open problem on the maximum possible
+chromatic number of a graph with thickness 2, or equivalently, bigenus 0. At present, it is
+known that this value is 9, 10, 11, or 12 (see [Get18]). The upper bound is derived from a
+standard coloring argument based on average degree, which Heawood [Hea90] also uses to
+color graphs embedded in arbitrary orientable surfaces. Heawood’s conjecture that his upper
+bound is tight is now called the Map Color Theorem [Rin74], proven by Ringel, Youngs, et
+al.
+Jackson and Ringel [JR00] conjecture a similar result for graphs biembeddable in higher-
+genus orientable surfaces. The maximum chromatic number over all graphs biembeddable
+in the surface Sg is called the bichromatic number of Sg and is denoted by χ2(Sg). The same
+coloring argument is used to prove the following Heawood-like inequality:
+Proposition 1.1 (Jackson and Ringel [JR00]). The bichromatic number of the orientable
+surface Sg, where g ≥ 1, is at most
+χ2(Sg) ≤
+�13 + √73 + 96g
+2
+�
+.
+1
+arXiv:2301.00286v1  [math.CO]  31 Dec 2022
+
+Conjecture 1.2 (Jackson and Ringel [JR00]). For all g ≥ 1, the bound in Proposition 1.1
+is tight.
+Just like the Map Color Theorem, this generalization of the Earth-Moon problem hardly
+resembles the original problem on the sphere: for all other surfaces, one might expect that
+the upper bound is always matched by a biembedding of a complete graph on the same
+number of vertices. Conjecture 1.2 thus has a stronger “graph-centric” formulation in terms
+of bigenus:
+Proposition 1.3 (Cabaniss and Jackson [CJ90]). The bigenus of the complete graph Kn is
+at least
+β(Kn) ≥
+�n2 − 13n + 24
+24
+�
+.
+Conjecture 1.4 (Cabaniss and Jackson [CJ90]). For all n ≥ 11,
+β(Kn) =
+�n2 − 13n + 24
+24
+�
+.
+The bigenus of the complete graph Kn can equal exactly n2 − 13n + 24/24 only when
+both embeddings of the biembedding are triangular. These so-called triangular biembeddings
+are only possible when n ≡ 0, 13, 16, 21 (mod 24), otherwise the expression is not an integer.
+With the exception of some small cases (β(Kn) is known for all n ≤ 14 [Rin59, BHK62,
+Tut63, Rin65, Bei69]), all other known constructions of minimum genus biembeddings of Kn
+have been triangular biembeddings. The second author [Sun22] found triangular embeddings
+of self-complementary graphs on 16, 21, and 24 vertices through computer search. One of the
+aforementioned residues, n ≡ 13 (mod 24), has been solved using current graphs, a covering
+space construction that has proven to be effective for finding triangular embeddings of dense
+graphs. The application of current graphs to biembeddings was initiated by Anderson and
+White [AW78], who found a pair of current graphs that produce a triangular biembedding
+of K37. Cabaniss and Jackson [CJ90] then solved the bigenus of K61 and K85. Finally, the
+second author [Sun22] completed this line of work by finding an infinite family of current
+graphs that produce triangular biembeddings of the complete graphs on n = 24s+13 vertices,
+for all s ≥ 1.
+The aforementioned current graphs are all of index 1, i.e., they are all 1-face embeddings.
+We solve another one of the residues by constructing triangular biembeddings of the complete
+graphs K24s+21, for all s ≥ 0, using index 3 current graphs.
+2
+Graph embeddings
+We assume prior knowledge of topological graph theory and the theory of current graphs. For
+background on these topics, see Gross and Tucker [GT87] and Ringel [Rin74]. In particular,
+2
+
+Section 9 of Ringel [Rin74] describes current graph constructions similar to the ones we will
+present here. For more information on the thickness parameter and its variants, see Beineke
+[Bei97].
+A cellular embedding of a graph G = (V, E) in the surface Sg is an injective mapping
+φ: G → Sg, where the components of Sg \ φ(G) are open disks. We call these disks faces. In
+this paper, all graph embeddings are cellular and in orientable surfaces. If the set of faces is
+denoted by F(φ), then its size is determined by the Euler polyhedral equation
+|V | − |E| + |F(φ)| = 2 − 2g.
+When G is simple, the Euler polyhedral equation implies a well-known inequality on the
+number of edges in G:
+Proposition 2.1. If G = (V, E) is a simple graph embedded in the orientable surface Sg,
+then
+|E| ≤ 3|V | − 6 + 6g,
+with equality if and only if the embedding is triangular.
+For biembeddings, a graph can have twice as many edges, and one can use this inequality
+to prove Propositions 1.1 and 1.3.
+To describe a cellular embedding combinatorially, each edge e ∈ E induces two arcs e+
+and e− with the same endpoints, each representing the two different directions in which
+e can be traversed. The set of such arcs is denoted E+. A rotation of a vertex is a cyclic
+permutation of the arcs leaving that vertex, and a rotation system of a graph is an assignment
+of a rotation to each vertex. When a graph is simple, it is sufficient to describe a rotation as
+a cyclic permutation of the vertex’s neighbors. The Heffter-Edmonds principle states that
+rotation systems are in one-to-one correspondence with cellular embeddings in orientable
+surfaces (see Section 3.2 of Gross and Tucker [GT87]). From a rotation system, a cellular
+embedding can be found through face-tracing, where each face-boundary walk corresponds
+to a cyclic sequence of arcs (e±
+1 , e±
+2 , . . . , e±
+i ).
+3
+Current graphs
+A current graph is an arc-labeled, embedded graph where the arc-labeling α : E+ → Zn \{0}
+satisfies α(e+) = −α(e−) for each edge e. We call Zn the current group and the arc labels
+currents. The index of a current graph is the number of faces in the embedding. Our current
+graphs are of index 3, and its face-boundary walks, which we call circuits, are labeled [0], [1],
+and [2]. Given a circuit, the log of the circuit replaces each arc with its current. We require
+that our current graphs satisfy a standard set of properties:
+(E1) The current graph has index 3.
+(E2) Each vertex has degree 3 and satisfies KCL.
+(E3) Each nonzero element of the current group Z3m appears at most once in the log of each
+circuit.
+3
+
+[0]
+[0]
+[1]
+[2]
+1
+1
+10
+19
+9
+9
+10
+10
+16
+13
+3
+3
+7
+7
+6
+1
+1
+A
+B
+A
+B
+[0]
+[0]
+[2]
+[1]
+4
+4
+9
+5
+5
+3
+2
+2
+7
+1
+6
+6
+8
+8
+4
+4
+4
+C
+D
+D
+C
+Figure 1: A pair of current graphs over Z21.
+(E4) If circuit [a] traverses arc e+ and circuit [b] traverses arc e−, then α(e+) ≡ b − a
+(mod 3).
+The derived embedding of a current graph satisfying the above properties is constructed
+in the following way: the vertex set is the current group Z3m, and the rotation at any vertex
+i ∈ Z3m (and hence its set of neighbors) is found by taking the log of circuit [i mod 3] and
+adding i (modulo Z3m) to each element. A vertex i is called a [k]-vertex if i mod 3 = k, i.e.,
+it is a vertex whose rotation is determined by circuit [k].
+Since every vertex has degree 3 and satisfies KCL, the derived embedding is triangular.
+Its genus thus has a simple formula:
+Proposition 3.1. Given an index 3 current graph, if the number of vertices is v, the current
+group is Z3m, and the derived embedding is connected, then its genus is (v − 6)m/4 + 1.
+Proof. Since there are three circuits and every vertex has degree 3, the average length of
+a circuit, and hence the average degree of the graph, is v. The above formula results from
+substituting E = 3mv/2 and V = 3m into Proposition 2.1.
+Our current graphs come in pairs, and each pair satisfies two additional properties:
+(E5) For each k = 0, 1, 2, each nonzero element of Z3m appears in the log of circuit [k] in
+exactly one of the two current graphs.
+(E6) Both current graphs have the same number of vertices.
+When these properties are satisfied, each possible edge between distinct vertices appears
+in exactly one of the two derived embeddings and by Proposition 3.1, the derived embeddings
+are on surfaces of the same genus. Consequently, we have a triangular biembedding of the
+complete graph K3m.
+4
+
+[0]
+[0]
+[1]
+[2]
+1
+1
+12s+9
+12s+9
+12s+10
+12s+10
+12s+6
+4
+4
+...
+...
+3s+1
+3s+1
+6s+9
+6s+9
+9s+10
+9s+10
+3
+3
+9s+7
+9s+7
+6s+3
+6s+3
+3s+4
+3s+4
+6s
+9s+4
+9s+4
+. ..
+. ..
+6s+1
+6s+1
+6
+6s+7
+6s+7
+6s+6
+1
+1
+A
+B
+A
+B
+[0]
+[0]
+[2]
+[1]
+6s+4
+6s+4
+12s+9
+6s+5
+6s+5
+3
+6s+2
+6s+2
+6
+6
+6s+8
+6s+8
+...
+...
+2
+2
+12s+6
+12s+6
+12s+8
+12s+8
+6s+4
+6s+4
+6s+4
+C
+D
+D
+C
+arithmetic
+arithmetic
+arithmetic
+Figure 2: Pairs of current graphs for all s ≥ 0 with current group Z24s+21.
+18s−3j+13
+18s+3j+16
+6j+3
+6j+3
+18s+3j+16
+18s−3j+13
+6j+3
+6j+3
+6s+3k+7
+6s−3k+1
+6k+6
+6k+6
+Figure 3: Current assignments on circular arcs.
+The two current graphs in Figure 1 satisfy properties (E1)–(E6). Hence, their derived
+embeddings form a triangular biembedding of K21. These current graphs contain frequently
+used elements in index 3 constructions that were first described in detail by Youngs [You70].
+The underlying graphs are (circular or M¨obius) ladders containing rungs. The rungs come
+in two varieties: simple rungs that are just vertical edges, and ring-shaped rungs, which have
+two more vertices connected by two parallel edges.
+4
+The main construction
+The current graphs in Figure 1 constitute the smallest instance of an infinite family:
+Theorem 4.1. The complete graph K24s+21 has a triangular biembedding for all s ≥ 0.
+Proof. The current graphs described in Figure 2 satisfy properties (E1)–(E6) and thus gen-
+erate triangular biembeddings of the complete graphs K24s+21, for all s ≥ 0. The sections
+labeled “arithmetic” describe part of the ladder where:
+• the rungs alternate between simple and ring-shaped,
+5
+
+• the vertical arcs alternate in direction, and
+• the currents on those vertical arcs form an arithmetic sequence with step size 3.
+In the interest of space, the labels on the circular arcs are given separately in Figure 3, where
+the variables have the ranges j = 0, . . . , 2s + 1 and k = 0, . . . , 2s. To check that the derived
+embeddings partition the edges of K24s+21, we categorize the edges based on their incident
+circuits. The horizontal edges are where circuit [0] meets with either circuit [1] or [2]; the
+simple rungs are where circuit [0] meets with itself; the vertical edges of ring-shaped rungs
+are where circuits [1] and [2] meet with themselves; and the circular arcs are where circuits
+[1] and [2] meet. One can use this information to check that property (E5) is satisfied.
+In both current graphs, there is at least one edge incident with circuits [0] and [1], and
+at least one edge incident with circuits [0] and [2]. Because of the presence of an arc with
+current 3, the derived embeddings of the first and second current graphs have a cycle passing
+through all the [1]-vertices and [0]-vertices, respectively. These two properties imply that
+the derived embeddings are connected.
+5
+Biembeddings on different surfaces
+Rearranging parts of the above infinite families of current graphs results in biembeddings
+into two surfaces of different genus. Cabaniss and Jackson [CJ90] say that a graph is (g, h)-
+biembeddable if it has an edge decomposition into two subgraphs, one of which is embeddable
+in the surface Sg, and the other in Sh.
+For each pair of current graphs in our main construction, each multiple of 3 corresponds
+to two rungs: in one graph, it appears as a current on a simple rung, and in the other
+graph, it appears twice on the vertical arcs of a ring-shaped rung. These two rungs can be
+swapped (possibly with some changes in arc directions) while preserving properties (E2)–
+(E5). Such exchanges have appeared in other constructions of index 3 current graphs (see,
+e.g., [JR80, Sun20]), except in those cases, they were rungs in the same current graph. In our
+situation, two pairs of rungs need to be swapped at the same time to ensure property (E1),
+that the indices of both current graphs stay at 3. Property (E6) is violated intentionally to
+get derived embeddings on different surfaces.
+Theorem 5.1. The complete graph K24s+21 is
+(b(s) − (8s + 7)k, b(s) + (8s + 7)k)-biembeddable,
+where b(s) = β(K24s+21) = 24s2 + 29s + 8 and k = 0, . . . , s + 1.
+Proof. Switching two normal rungs with two ring-shaped rungs changes the total number
+of vertices in both current graphs by 4. From Proposition 3.1, the genus must increase or
+decrease by 8s + 7. The first current graph has 2s + 2 ring-shaped rungs (the second current
+graph has one fewer), so up to s+1 pairs of rungs can be exchanged. Finally, the connectivity
+argument at the end of the proof of Theorem 4.1 is still valid even if the rungs with current
+3 are swapped.
+6
+
+[0]
+[0]
+[1]
+[2]
+1
+1
+9
+10
+10
+3
+7
+7
+6
+1
+1
+A
+B
+A
+B
+[0]
+[0]
+[2]
+[1]
+4
+4
+19
+10
+9
+9
+5
+5
+13
+16
+3
+3
+2
+2
+7
+1
+6
+6
+8
+8
+4
+4
+4
+C
+D
+D
+C
+Figure 4: K21 is (1, 15)-biembeddable.
+Figure 4 shows a swap on the two current graphs that originally appeared in Figure 1.
+Plugging in s = 0 and k = 1 into Theorem 5.1 shows that the derived embeddings of the
+graphs are on the torus and the genus 15 surface.
+References
+[AW78]
+Ian Anderson and Arthur T White. Current graphs and bi-embeddings. Journal
+of Graph Theory, 2(3):231–239, 1978.
+[Bei69]
+Lowell W. Beineke. Minimal decompositions of complete graphs into subgraphs
+with embeddability properties. Canadian Journal of Mathematics, 21:992–1000,
+1969.
+[Bei97]
+Lowell W. Beineke. Biplanar graphs: A survey. Computers & Mathematics with
+Applications, 34(11):1–8, 1997.
+[BHK62] Joseph Battle, Frank Harary, and Yukihiro Kodama. Every planar graph with
+nine points has a nonplanar complement. Bulletin of the American Mathematical
+Society, 68(6):569–571, 1962.
+[CJ90]
+Sharon Cabaniss and Bradley W. Jackson.
+Infinite families of bi-embeddings.
+Discrete Mathematics, 82(2):127–141, 1990.
+[Get18]
+Ellen Gethner. To the Moon and Beyond. In Graph Theory—Favorite Conjectures
+and Open Problems – 2, pages 115–133. Springer, 2018.
+[GT87]
+Jonathan L. Gross and Thomas W. Tucker. Topological Graph Theory. Wiley &
+Sons, 1987.
+7
+
+[Hea90]
+Percy John Heawood. Map Colour Theorem. Quarterly Journal of Mathematics,
+24:332–338, 1890.
+[JR80]
+Mark Jungerman and Gerhard Ringel. Minimal triangulations on orientable sur-
+faces. Acta Mathematica, 145(1):121–154, 1980.
+[JR00]
+Brad Jackson and Gerhard Ringel. Variations on Ringel’s earth-moon problem.
+Discrete Mathematics, 211(1-3):233–242, 2000.
+[Rin59]
+Gerhard Ringel. F¨arbungsprobleme auf fl¨achen und graphen. Deutscher Verlag der
+Wissenschaften, 1959.
+[Rin65]
+Gerhard Ringel. Die toroidale dicke des vollst¨andigen graphen. Mathematische
+Zeitschrift, 87(1):19–26, 1965.
+[Rin74]
+Gerhard Ringel. Map Color Theorem. Springer Science & Business Media, 1974.
+[Sun20]
+Timothy Sun.
+Simultaneous current graph constructions for minimum trian-
+gulations and complete graph embeddings.
+Ars Mathematica Contemporanea,
+18(2):309–337, 2020.
+[Sun22]
+Timothy Sun. On the bigenus of the complete graphs. Australasian Journal of
+Combinatorics, 81(1):212–219, 2022.
+[Tut63]
+William T. Tutte. The non-biplanar character of the complete 9-graph. Canadian
+Mathematical Bulletin, 6(3):319–330, 1963.
+[You70]
+J.W.T. Youngs. Solution of the Heawood map-coloring problem — Cases 3, 5, 6,
+and 9. Journal of Combinatorial Theory, 8(2):175–219, 1970.
+8
+

6NAyT4oBgHgl3EQfcffh/content/tmp_files/load_file.txt

@@ -0,0 +1,226 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf,len=225
+page_content='Index 3 biembeddings of the complete graphs  Juvenal F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Barajas and Timothy Sun  Department of Computer Science  San Francisco State University  Abstract  We show that the complete graphs on 24s + 21 vertices have decompositions into  two edge-disjoint subgraphs, each of which triangulates an orientable surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  The  special case where the two surfaces are homeomorphic solves a generalized Earth-  Moon problem for that surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Unlike previous constructions, these pairs of triangular  embeddings are derived from index 3 current graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  1  Introduction  There are many graph parameters that generalize the notion of planarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Perhaps the most  well-known of such parameters is the genus of the graph, which is the smallest value g such  that the graph has an embedding in Sg, the orientable surface of genus g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' A less-studied  parameter is the thickness of a graph, which is the size of the smallest partition of the edges  into planar subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' A graph is said to be biembeddable in surfaces S and S′ if it can  be decomposed into two edge-disjoint subgraphs, one of which embeds in S and the other  embeds in S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' When S is homeomorphic to S′, we simply say that the graph is biembeddable  in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' We consider a variant of both genus and thickness, the bigenus of a graph β(G), which  is defined to be the smallest value g such that the graph G is biembeddable in Sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  The Earth-Moon problem is a longstanding open problem on the maximum possible  chromatic number of a graph with thickness 2, or equivalently, bigenus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' At present, it is  known that this value is 9, 10, 11, or 12 (see [Get18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' The upper bound is derived from a  standard coloring argument based on average degree, which Heawood [Hea90] also uses to  color graphs embedded in arbitrary orientable surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Heawood’s conjecture that his upper  bound is tight is now called the Map Color Theorem [Rin74], proven by Ringel, Youngs, et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  Jackson and Ringel [JR00] conjecture a similar result for graphs biembeddable in higher-  genus orientable surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' The maximum chromatic number over all graphs biembeddable  in the surface Sg is called the bichromatic number of Sg and is denoted by χ2(Sg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' The same  coloring argument is used to prove the following Heawood-like inequality:  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='1 (Jackson and Ringel [JR00]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' The bichromatic number of the orientable  surface Sg, where g ≥ 1, is at most  χ2(Sg) ≤  �13 + √73 + 96g  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='00286v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='CO]  31 Dec 2022  Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='2 (Jackson and Ringel [JR00]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' For all g ≥ 1, the bound in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='1  is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  Just like the Map Color Theorem, this generalization of the Earth-Moon problem hardly  resembles the original problem on the sphere: for all other surfaces, one might expect that  the upper bound is always matched by a biembedding of a complete graph on the same  number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='2 thus has a stronger “graph-centric” formulation in terms  of bigenus:  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='3 (Cabaniss and Jackson [CJ90]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' The bigenus of the complete graph Kn is  at least  β(Kn) ≥  �n2 − 13n + 24  24  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='4 (Cabaniss and Jackson [CJ90]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' For all n ≥ 11,  β(Kn) =  �n2 − 13n + 24  24  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  The bigenus of the complete graph Kn can equal exactly n2 − 13n + 24/24 only when  both embeddings of the biembedding are triangular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' These so-called triangular biembeddings  are only possible when n ≡ 0, 13, 16, 21 (mod 24), otherwise the expression is not an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  With the exception of some small cases (β(Kn) is known for all n ≤ 14 [Rin59, BHK62,  Tut63, Rin65, Bei69]), all other known constructions of minimum genus biembeddings of Kn  have been triangular biembeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' The second author [Sun22] found triangular embeddings  of self-complementary graphs on 16, 21, and 24 vertices through computer search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' One of the  aforementioned residues, n ≡ 13 (mod 24), has been solved using current graphs, a covering  space construction that has proven to be effective for finding triangular embeddings of dense  graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' The application of current graphs to biembeddings was initiated by Anderson and  White [AW78], who found a pair of current graphs that produce a triangular biembedding  of K37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Cabaniss and Jackson [CJ90] then solved the bigenus of K61 and K85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Finally, the  second author [Sun22] completed this line of work by finding an infinite family of current  graphs that produce triangular biembeddings of the complete graphs on n = 24s+13 vertices,  for all s ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  The aforementioned current graphs are all of index 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=', they are all 1-face embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  We solve another one of the residues by constructing triangular biembeddings of the complete  graphs K24s+21, for all s ≥ 0, using index 3 current graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  2  Graph embeddings  We assume prior knowledge of topological graph theory and the theory of current graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' For  background on these topics, see Gross and Tucker [GT87] and Ringel [Rin74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' In particular,  2  Section 9 of Ringel [Rin74] describes current graph constructions similar to the ones we will  present here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' For more information on the thickness parameter and its variants, see Beineke  [Bei97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  A cellular embedding of a graph G = (V, E) in the surface Sg is an injective mapping  φ: G → Sg, where the components of Sg \\ φ(G) are open disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' We call these disks faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' In  this paper, all graph embeddings are cellular and in orientable surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' If the set of faces is  denoted by F(φ), then its size is determined by the Euler polyhedral equation  |V | − |E| + |F(φ)| = 2 − 2g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  When G is simple, the Euler polyhedral equation implies a well-known inequality on the  number of edges in G:  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' If G = (V, E) is a simple graph embedded in the orientable surface Sg,  then  |E| ≤ 3|V | − 6 + 6g,  with equality if and only if the embedding is triangular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  For biembeddings, a graph can have twice as many edges, and one can use this inequality  to prove Propositions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  To describe a cellular embedding combinatorially, each edge e ∈ E induces two arcs e+  and e− with the same endpoints, each representing the two different directions in which  e can be traversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' The set of such arcs is denoted E+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' A rotation of a vertex is a cyclic  permutation of the arcs leaving that vertex, and a rotation system of a graph is an assignment  of a rotation to each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' When a graph is simple, it is sufficient to describe a rotation as  a cyclic permutation of the vertex’s neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' The Heffter-Edmonds principle states that  rotation systems are in one-to-one correspondence with cellular embeddings in orientable  surfaces (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='2 of Gross and Tucker [GT87]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' From a rotation system, a cellular  embedding can be found through face-tracing, where each face-boundary walk corresponds  to a cyclic sequence of arcs (e±  1 , e±  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' , e±  i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  3  Current graphs  A current graph is an arc-labeled, embedded graph where the arc-labeling α : E+ → Zn \\{0}  satisfies α(e+) = −α(e−) for each edge e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' We call Zn the current group and the arc labels  currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' The index of a current graph is the number of faces in the embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Our current  graphs are of index 3, and its face-boundary walks, which we call circuits, are labeled [0], [1],  and [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Given a circuit, the log of the circuit replaces each arc with its current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' We require  that our current graphs satisfy a standard set of properties:  (E1) The current graph has index 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  (E2) Each vertex has degree 3 and satisfies KCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  (E3) Each nonzero element of the current group Z3m appears at most once in the log of each  circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  3  [0]  [0]  [1]  [2]  1  1  10  19  9  9  10  10  16  13  3  3  7  7  6  1  1  A  B  A  B  [0]  [0]  [2]  [1]  4  4  9  5  5  3  2  2  7  1  6  6  8  8  4  4  4  C  D  D  C  Figure 1: A pair of current graphs over Z21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  (E4) If circuit [a] traverses arc e+ and circuit [b] traverses arc e−, then α(e+) ≡ b − a  (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  The derived embedding of a current graph satisfying the above properties is constructed  in the following way: the vertex set is the current group Z3m, and the rotation at any vertex  i ∈ Z3m (and hence its set of neighbors) is found by taking the log of circuit [i mod 3] and  adding i (modulo Z3m) to each element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' A vertex i is called a [k]-vertex if i mod 3 = k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=',  it is a vertex whose rotation is determined by circuit [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  Since every vertex has degree 3 and satisfies KCL, the derived embedding is triangular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  Its genus thus has a simple formula:  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Given an index 3 current graph, if the number of vertices is v, the current  group is Z3m, and the derived embedding is connected, then its genus is (v − 6)m/4 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Since there are three circuits and every vertex has degree 3, the average length of  a circuit, and hence the average degree of the graph, is v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' The above formula results from  substituting E = 3mv/2 and V = 3m into Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  Our current graphs come in pairs, and each pair satisfies two additional properties:  (E5) For each k = 0, 1, 2, each nonzero element of Z3m appears in the log of circuit [k] in  exactly one of the two current graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  (E6) Both current graphs have the same number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  When these properties are satisfied, each possible edge between distinct vertices appears  in exactly one of the two derived embeddings and by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='1, the derived embeddings  are on surfaces of the same genus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Consequently, we have a triangular biembedding of the  complete graph K3m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  4  [0]  [0]  [1]  [2]  1  1  12s+9  12s+9  12s+10  12s+10  12s+6  4  4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  3s+1  3s+1  6s+9  6s+9  9s+10  9s+10  3  3  9s+7  9s+7  6s+3  6s+3  3s+4  3s+4  6s  9s+4  9s+4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='.  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='.  6s+1  6s+1  6  6s+7  6s+7  6s+6  1  1  A  B  A  B  [0]  [0]  [2]  [1]  6s+4  6s+4  12s+9  6s+5  6s+5  3  6s+2  6s+2  6  6  6s+8  6s+8  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  2  2  12s+6  12s+6  12s+8  12s+8  6s+4  6s+4  6s+4  C  D  D  C  arithmetic  arithmetic  arithmetic  Figure 2: Pairs of current graphs for all s ≥ 0 with current group Z24s+21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  18s−3j+13  18s+3j+16  6j+3  6j+3  18s+3j+16  18s−3j+13  6j+3  6j+3  6s+3k+7  6s−3k+1  6k+6  6k+6  Figure 3: Current assignments on circular arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  The two current graphs in Figure 1 satisfy properties (E1)–(E6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Hence, their derived  embeddings form a triangular biembedding of K21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' These current graphs contain frequently  used elements in index 3 constructions that were first described in detail by Youngs [You70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  The underlying graphs are (circular or M¨obius) ladders containing rungs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' The rungs come  in two varieties: simple rungs that are just vertical edges, and ring-shaped rungs, which have  two more vertices connected by two parallel edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  4  The main construction  The current graphs in Figure 1 constitute the smallest instance of an infinite family:  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' The complete graph K24s+21 has a triangular biembedding for all s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' The current graphs described in Figure 2 satisfy properties (E1)–(E6) and thus gen-  erate triangular biembeddings of the complete graphs K24s+21, for all s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' The sections  labeled “arithmetic” describe part of the ladder where:  the rungs alternate between simple and ring-shaped,  5  the vertical arcs alternate in direction, and  the currents on those vertical arcs form an arithmetic sequence with step size 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  In the interest of space, the labels on the circular arcs are given separately in Figure 3, where  the variables have the ranges j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' , 2s + 1 and k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' , 2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' To check that the derived  embeddings partition the edges of K24s+21, we categorize the edges based on their incident  circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' The horizontal edges are where circuit [0] meets with either circuit [1] or [2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' the  simple rungs are where circuit [0] meets with itself;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' the vertical edges of ring-shaped rungs  are where circuits [1] and [2] meet with themselves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' and the circular arcs are where circuits  [1] and [2] meet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' One can use this information to check that property (E5) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  In both current graphs, there is at least one edge incident with circuits [0] and [1], and  at least one edge incident with circuits [0] and [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Because of the presence of an arc with  current 3, the derived embeddings of the first and second current graphs have a cycle passing  through all the [1]-vertices and [0]-vertices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' These two properties imply that  the derived embeddings are connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  5  Biembeddings on different surfaces  Rearranging parts of the above infinite families of current graphs results in biembeddings  into two surfaces of different genus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Cabaniss and Jackson [CJ90] say that a graph is (g, h)-  biembeddable if it has an edge decomposition into two subgraphs, one of which is embeddable  in the surface Sg, and the other in Sh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  For each pair of current graphs in our main construction, each multiple of 3 corresponds  to two rungs: in one graph, it appears as a current on a simple rung, and in the other  graph, it appears twice on the vertical arcs of a ring-shaped rung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' These two rungs can be  swapped (possibly with some changes in arc directions) while preserving properties (E2)–  (E5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Such exchanges have appeared in other constructions of index 3 current graphs (see,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=', [JR80, Sun20]), except in those cases, they were rungs in the same current graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' In our  situation, two pairs of rungs need to be swapped at the same time to ensure property (E1),  that the indices of both current graphs stay at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Property (E6) is violated intentionally to  get derived embeddings on different surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' The complete graph K24s+21 is  (b(s) − (8s + 7)k, b(s) + (8s + 7)k)-biembeddable,  where b(s) = β(K24s+21) = 24s2 + 29s + 8 and k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' , s + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Switching two normal rungs with two ring-shaped rungs changes the total number  of vertices in both current graphs by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='1, the genus must increase or  decrease by 8s + 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' The first current graph has 2s + 2 ring-shaped rungs (the second current  graph has one fewer), so up to s+1 pairs of rungs can be exchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Finally, the connectivity  argument at the end of the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='1 is still valid even if the rungs with current  3 are swapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  6  [0]  [0]  [1]  [2]  1  1  9  10  10  3  7  7  6  1  1  A  B  A  B  [0]  [0]  [2]  [1]  4  4  19  10  9  9  5  5  13  16  3  3  2  2  7  1  6  6  8  8  4  4  4  C  D  D  C  Figure 4: K21 is (1, 15)-biembeddable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  Figure 4 shows a swap on the two current graphs that originally appeared in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  Plugging in s = 0 and k = 1 into Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='1 shows that the derived embeddings of the  graphs are on the torus and the genus 15 surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  References  [AW78]  Ian Anderson and Arthur T White.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Current graphs and bi-embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Journal  of Graph Theory, 2(3):231–239, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  [Bei69]  Lowell W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Beineke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Minimal decompositions of complete graphs into subgraphs  with embeddability properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Canadian Journal of Mathematics, 21:992–1000,  1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  [Bei97]  Lowell W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Beineke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Biplanar graphs: A survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Computers & Mathematics with  Applications, 34(11):1–8, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  [BHK62] Joseph Battle, Frank Harary, and Yukihiro Kodama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Every planar graph with  nine points has a nonplanar complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Bulletin of the American Mathematical  Society, 68(6):569–571, 1962.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  [CJ90]  Sharon Cabaniss and Bradley W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
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+page_content=' Springer, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
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+page_content=' Map Colour Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Quarterly Journal of Mathematics,  24:332–338, 1890.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  [JR80]  Mark Jungerman and Gerhard Ringel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content=' Minimal triangulations on orientable sur-  faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
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+page_content=' Variations on Ringel’s earth-moon problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
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+page_content=' Springer Science & Business Media, 1974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
+page_content='  [Sun20]  Timothy Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'}
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+Wavenumber Scattering and Inter-band Targeted Energy Transfer in Phononic
+Lattices with Local Vibro-Impact Nonlinearities
+Joshua R. Tempelman, Alexander F. Vakakis, and Kathryn H. Matlack
+Department of Mechanical Science and Engineering,
+University of Illinois at Urbana-Champaign, 1206 W Green St, Urbana, IL 61801
+We propose a method for manipulating wave propagation in phononic lattices by employing local
+vibro-impact (VI) nonlinearities to scatter energy across the underling linear band structure of the
+lattice, and transfer energy from lower to higher optical bands. Inspired by recent developments
+in the field of nonlinear targeted energy transfer (TET) using non-resonant energy exchanges, we
+achieve this using spatially localized VI forces that redistribute energy across the linear spectrum
+of the lattice in a non-resonant fashion. First, a 1-dimensional (1D), 2-band phononic lattice with
+embedded VI unit cells is computationally studied to demonstrate that energy is scattered in the
+wavenumber domain, and this nonlinear scattering mechanism depends on the energy of the propa-
+gating wave. Next, a 4-band lattice is studied with a similar technique to demonstrate the concept
+of inter-band targeted energy transfer (IBTET) and to establish analogous scaling relations with
+respect to energy. To interpret the results of IBTET, we study the nonlinear normal modes (NNMs)
+of a reduced order model (ROM) of the VI unit cell in the 4-band lattice, using the method of
+numerical continuation. Interestingly, the slope of the frequency-energy branches of the ROM cor-
+responding to the 1:1 resonance NNM matches remarkably well with the dependence of IBTET
+to input energy in the 4-band lattice. In both phononic lattices, it is shown that there exists a
+maximum energy transfer at moderate input energies, followed by a power law decay of relative
+energy transfer either to the wavenumber domain or between bands on input energy; this power law
+dependence is additionally validated by the ROM. Moreover, relations between the dynamics of the
+VI lattice and the NNMs of the underlying Hamiltonian system provide physical interpretations for
+the relative energy transfers. Hence, we present a predictive framework to computationally explore
+non-resonant energy transfers across the linear band structure of phononic lattices with local strong
+non-smooth nonlinearities and provide a comprehensive physics-based interpretation of these energy
+transfers based on the nonlinear dynamics of the lower-dimensional ROM.
+I.
+INTRODUCTION
+Periodicity has been leveraged to control acoustic and
+elastic energy propagation in linear time-invariant (LTI)
+phononic metamaterials [1–3]. Such systems are typically
+designed on a unit cell level whereby the application of
+the Bloch theorem allows one to engineer a linear band
+structure which can enable or augment specified wave
+phenomena with diverse applications such as lensing [4],
+energy harvesting [5–7], vibration isolation [8–10], wave
+steering [11], mechanical logic circuits [12], mechanical
+signal processing [13], and topological insulation [14–16].
+For LTI phononic systems, a propagating wave remains
+stationary on a prescribed subset of its band structure,
+and is invariant to amplitude (or energy) as the dynam-
+ics are completely described by the superposition prin-
+ciple [3]. However, it is often desirable to predictively
+tune wave propagation in phononic materials such that
+the propagating wave shifts to a different subset of its
+band structure. To this end, one must either manipu-
+late the underlying band structure altogether by utiliz-
+ing external forces or nonlinearity [3, 17], or find methods
+to modify the distribution of (or, equivalently, passively
+manage) energy across a fixed underlying band structure.
+Whereas band manipulation has been achieved by in-
+troducing e.g., electromagnetic, magnetic, mechanical, or
+thermal fields [18–23], nonlinear mechanisms offer the key
+advantage of being passive and tunable (self-adaptive)
+to energy, frequency and wavenumber content [17, 24].
+For instance, the effective dispersion relations of granular
+chains with Hertzian contact laws are tunable by locally
+linearizing about various pre-compression states [25–27].
+Moreover, passive nonlinear mechanisms posses intrinsic
+frequency-amplitude dependencies, and the correspond-
+ing shifts to the band structures can be described by
+perturbations of the underlying linearized band struc-
+ture [28] for low energy or by the nonlinear normal modes
+(NNMs) at high energy [29–32]. Aside from band struc-
+ture manipulation, distributed nonlinearity in periodic
+chains has enabled exotic wave behavior in lattices with
+no properly defined band structure such as stegetons [33],
+solitons [34], and breathers [35, 36].
+Herein, we aim to develop mechanisms to manipulate
+propagating energy in phononic metamaterials using lo-
+calized nonlinearities to transfer energy across the un-
+derlying linear band structure. In the absence of external
+actions, the transfer of energy across an underlying linear
+spectrum requires a nonlinear mechanism which has the
+capability to transfer energy form one modal subspace
+to another. Such a mechanism is fundamental to achiev-
+ing targeted energy transfer (TET), a concept which has
+been rigorously studied by the nonlinear dynamics com-
+munity from the point of view of nonlinear modal dynam-
+ics [37]. TET is most commonly achieved by employing
+localized nonlinear energy sinks (NESs) which alter the
+global dynamics of a primary linear structure to which
+they are attached, with typical applications in vibration
+arXiv:2301.05302v1  [physics.app-ph]  12 Jan 2023
+
+2
+mitigation [38–52]. The TET phenomenon relies on res-
+onance capture of the NES to a resonance manifold, and
+thus traditional TET is intrinsically suited for systems
+with smooth nonlinearities and periodic excitations [37].
+However, theoretical and numerical support has recently
+been extended to systems with non-stationary dynam-
+ics [53] and systems with non-smooth nonlinearities such
+as idealized vibro-impact (VI) laws [54–56].
+The use of nonlinear attachments in acoustic wave
+guides (either bulk or periodic) have demonstrated un-
+precedented properties in acoustical systems [57].
+For
+instance, a small mass supported by an essential (non-
+linearizable) stiffness nonlinearity in parallel to a viscous
+damper attached to a periodic array of oscillators has
+been shown to host a rich variety of nonlinear dynamics
+when interacting with traveling waves [58], and are even
+capable of arresting incoming pulses [59]. Moreover, with
+the incorporation of hierarchical mass scales and asym-
+metry, similar systems have achieved nonreciprocity [60–
+62]. These effects have been extended for systems with lo-
+cal nonlinear gates that enable global non-reciprocity and
+effective diode-type features in both continuous waveg-
+uides [63] and discrete oscillator chains [64, 65]. In addi-
+tion to reciprocity, the concept of local gates in waveg-
+uides has recently been extended to produce effective
+mechanical filters for layered metamaterials with inter-
+faces [66] and for discrete periodic chains [67].
+Recently, new ideas have emerged in the area of TET
+which explore non-resonant energy exchanges in a di-
+rectly forced primary linear structure using VI nonlinear-
+ity to redistribute modal energy within its modal space,
+termed inter modal targeted energy transfer [68]. This
+methodology was studied computationally in [69] for a
+discrete mulit-DoF structure, and was later experimen-
+tally verified in [70] for the case of a cantilever beam un-
+dergoing VIs. Unlike resonant mechanisms, non-resonant
+mechanisms aim to scatter energy across the underlying
+linear modal basis in a low-to-high frequency fashion. In
+a similar fashion, Theurich et al. studied the directed
+scattering of energy to higher modes in a harmonically
+excited beam, and found that the effectiveness of the en-
+ergy scatter is dependent on the dynamic regimes of the
+VI system considered [71].
+To date, non-resonant energy scattering concepts have
+not been extended to periodic phononic metamaterials
+from a wave propagation perspective. The most notable
+differences between modal and periodic acoustical sys-
+tems is that the first employs a modal basis to describe
+stationary vibrations (and is suitable for systems of fi-
+nite extent whose dynamics are governed by slow time
+scales), while the latter a continuous band structure to
+describe propagating waves (and applies to unbounded /
+large-scale systems whose acoustics are governed by fast
+time scales). Hence, several natural questions arise when
+considering non-resonant TET phenomena in a phononic
+material. Namely, to what extent can the linear wave
+propagation be scattered in the wave number domain
+across a dispersion branch, and to what capacity can
+energy be irreversibly transferred from one band to an-
+other by use of localized VI nonlinearities. This paper
+addresses these questions with extensive computational
+probing, new post-processing techniques, and physics-
+based reasoning of the resulting nonlinear acoustic phe-
+nomena.
+We begin by studying the effects of VI nonlinearity in
+a 2-band phononic lattice of diatomic resonators by ex-
+tensive simulation and numerical post-processing of the
+acoustics. For this, we focus on the energy scattered of
+energy across the frequency/wavenumber domain of the
+single optical band of this lattice as a function of the
+number of local VI unit cells and as a function of the
+incident wave energy grows. Next, we consider a 4-band
+phononic lattice, which has one acoustic and three opti-
+cal bands over a relatively broad frequency/wavenumber
+range.
+This band structure, coupled with the strong
+VI nonlinearities, allows for low-to-high frequency en-
+ergy generation of the impacts, as well as targeted energy
+transfers across bands. This brings about the new non-
+linear acoustic phenomenon of inter-band targeted energy
+transfer (IBTET).
+Accordingly, the organization of this paper is as fol-
+lows.
+Section II provides a system description of the
+unit cell of the 2-band phononic lattice, a computational
+framework for studying wavenumber scattering within
+the single optical band induced by the VIs, and quan-
+tification of the spectral disorder generated by the VIs
+with respect to energy. Section III extends the study to a
+4-band phononic lattice and presents a method for trans-
+ferring energy from lower-to-higher optical bands via VIs,
+together with relationships between these transfers and
+the total system energy. Section IV presents a 2 DoF re-
+duced order model (ROM) which is studied through the
+from the perspective of NNM analysis in order to provide
+a physics-based understanding of the results of Sections II
+and III, and relate the nonlinear dynamics of the ROM to
+the IBTET occurring in the lattice. Lastly, Section V of-
+fers concluding remarks and some suggestions for further
+extension of this work.
+II.
+WAVENUMBER ENERGY SCATTERING
+We begin by studying a 1D phononic lattice in the form
+of a diatomic resonator chain and embed VI contact laws
+in select (local) resonators while preserving the global
+linear structure of the lattice.
+The system is compu-
+tationally explored by performing numerical simulations
+with wave packet excitations over an array of excitation
+amplitudes and wave numbers. The resulting data sets
+were next post-processed with a suite of discrete signal
+processing methods in the spatial-temporal domain to
+uncover the underlying trends of energy scattering in the
+wavenumber domain as the excitation level (input en-
+ergy) changes.
+
+3
+1 VI
+3 VI
+5 VI
+10 VI
+15 VI
+20 VI
+Configurations
+Unit Cell with VI
+Unit Cell Without VI
+Cell 1
+Cell 150
+Cell 300
+Cell 600
+Underlying Linear System
+Unit Cell With VI
+Unit Cell Without VI
+(c)
+`
+(a)
+(b)
+(d)
+FIG. 1.
+The linear phononic lattice composed of coupled
+(host) masses with embedded internal resonators which may
+or may not undergo vibro-impacts: (a) The primary linear pe-
+riodic system with the underlying linear dispersion relation.
+The nominal unit cell (b) without a VI nonlinearity and (c)
+with the VI nonlinearity; (d) schematics of finite lattice con-
+figurations which are comprised of the linear phononic lattice
+with various number of embedded VI unit cells.
+A.
+System Description and Simulations
+We consider a linear diatomic lattice constructed by
+the periodic tessellation of 1-D unit cells in the x-
+direction (Fig. 1(a)).
+Each unit cell is composed of a
+host mass and within it a resonator, which depending on
+the existence (absence) of rigid barriers it may (may not)
+experience vibro-impacts (see Fig. 1). The equations of
+motion for the k-th cell in the infinite phononic lattice
+are written as:
+m1¨uk
+1 = k1(uk−1
+1
++ uk+1
+1
+− 2xk
+1) + k2(uk
+2 − uk
+1),
+m2¨uk
+2 = k2(uk
+1 − uk
+2).
+(1)
+Imposing the Bloch ansatz, u(x) = ˜u exp(iκx − iωt), re-
+covers the linear dispersion derived from the underlying
+Bloch eigenvalue problem, ˜u( ˜Mω2− ˜K(κ)) = 0, where ˜M
+and ˜K are the Bloch-periodic mass and stiffness matrices
+of a unit cell. This yields two pass bands for this lattice,
+namely a lower-frequency acoustic band and a higher-
+frequency optical band. To computationally probe the
+effects of impact dynamics on the linear wave propaga-
+tion, we consider six different lattice configurations, each
+corresponding to a unique arrangement of VI unit cells
+embedded in the linear lattice with the number of VIs
+ranging between 1 and 20. To study the scattering of
+the input wave energy in the wavenumber domain accu-
+rately, a large finite system should be used for sufficient
+wavenumber resolution. To this end, we consider a finite
+configuration of 600 unit cells (1200 DoF) governed by
+M¨u + Ku + FNL(u, ˙u) = Fext(t)
+(2)
+where M and K are the finite mass and stiffness matri-
+ces, FNL(u, ˙u) the vector of nonlinear stiffness and vis-
+cous damping terms, and Fext(t) the vector of excita-
+tions. Excitation is provided in the form of a windowed
+harmonic function,
+Fk(t) =
+�
+W(t) sin(Ωt)
+k = 1
+0,
+otherwise
+(3)
+where W(t) = A
+�
+H(t) − H
+�
+t − 2πNcyc
+Ω
+�� �
+1 − cos
+�
+Ωt
+Ncyc
+��
+is a windowing function, H(t) the Heaviside function, A
+the forcing amplitude, Ncyc the number of cycles in the
+window, and Ω the center frequency of excitation. The
+nonlinear VI cells that are locally distributed through
+the lattice provide the following VI forces,
+FNL(wk) = kc
+�
+(wk − ∆i)n
++ − (−wk − ∆k)n
++
+�
+g( ˙wk, ˙w−
+k )
+(4)
+where wk(t) = uk
+2(t) − uk
+1(t) is the relative deflection be-
+tween the resonator and its host mass, n the nonlinearity
+coefficient which is set to n = 3/2 to emulate Hertzian
+contact unless otherwise stated, ∆k the clearance of the
+k-th VI in the lattice, and kc = 2EVI
+√RVI
+3(1−ν2)
+the stiffness
+parameter for Hertzian contacts, with EVI, RVI, and ν
+being the modulus, radius, and Poisson ratio of the VI,
+respectively. The notation ( )+ indicates that only pos-
+itive arguments are to be considered. We assume an in-
+elastic contact law as derived by Hunt and Crossly [72]
+which provides a hysteresis dissipation function derived
+from the work-energy principal in terms of the indenta-
+tion depth, g( ˙wk, ˙w−
+k ) =
+�
+1 − 3(1−r)
+2 ˙w−
+k
+˙wk
+�
+, where ˙w−
+k is
+the velocity ˙wk immediately before impact and r the co-
+efficient of restitution. Note that Eq (4) does not modify
+the underling linear band structure of the extended lat-
+tice. Moreover, for amplitudes such that wk < ∆k for
+each VI, the wave propagation remains completely linear
+as no VI experiences contact.
+Numerical simulations of equations (2) were carried
+out using the ODE78 routine in MATLAB. The center
+frequency of the excitation was selected based on the de-
+sired excitation wavenumbers, which were considered in
+the range between 2π/9 ≤ κ⋆ ≤ 7π/9 to ensure con-
+sistency in observations across the optical band struc-
+ture; however we focus only on κ⋆ = 5π/9 and refer the
+reader to supplemental material for additional results.
+The excitation frequencies were chosen within the op-
+tical band to ensure out-of-phase motion between each
+resonator and host mass and thus excite the VI (note
+in-phase motion, characteristic of the acoustic branch,
+will not excite the VI). Clearances were nominally set to
+range between 0.0002 and 0.0001 m with a logarithmic
+dependence on position from the leading VI unit cell to
+account for the momentum loss of the wave as it passes
+successively through VI cells in the lattice.
+The mass
+and stiffness of the linear resonator (i.e., in the absence
+of rigid barriers and VIs - cf. Fig. 1) were selected to em-
+ulate realistic resonator systems considered in the liter-
+ature [73]. Table I lists nominal parameters for stiffness,
+mass, and VI stiffness parameters. Within this frame-
+work, an ensemble of simulation data was constructed for
+25 logarithmically increasing forcing amplitudes for each
+
+4
+FIG. 2. Simulation results for a 5-VI configuration at excita-
+tion wavenumber k⋆ = 5π/9 (in the optical band of the linear
+lattice) with columns corresponding to (a) low, (b) medium,
+and (c) high amplitude excitations. For each amplitude, the
+rows depict (i)the spatio-temporal evolution of the kinetic en-
+ergy of the propagating wave, (ii) the temporal variation of
+the wavenumber distribution in the lattice, and (iii) the nu-
+merically computed dispersion computed using the entirety of
+the simulation with a gray dashed line superimposed to depict
+the analytical dispersion of the infinite liner lattice.
+configuration and excitation wavenumber considered.
+TABLE I. Parameters used for the di-atomic resonator chain
+m1
+[kg]
+m2
+[kg]
+k1
+[kN/m]
+k2
+[kN/m]
+ν
+r
+RVI
+[m]
+EVI
+[MPa]
+0.01
+0.08
+90
+90
+0.3
+0.7
+0.005 200
+B.
+Influence of VIs on Wave Propagation
+A suite of numerical post-processing tools were devel-
+oped to study the influence of the VIs on wave prop-
+agation in the lattice.
+The focus of the post pro-
+cessing was to uncover spectral content in the spatial
+and spatial-temporal domains with an emphasis on fre-
+quency/wavenumber scattering of the energy. This was
+achieved using Fourier and Wavelet transformations to
+study the energy content across the band structure in
+various domains including time, space, frequency, and
+wavenumber. In this section, we focus on a narrow sub-
+set of three simulations conducted at low, medium and
+high forcing amplitudes in order to build intuition on the
+post-processing analysis procedures and a qualitative de-
+pendence on system energy. Quantitative results across
+FIG. 3. The spatial wavelet transformations of the propagat-
+ing waves considered in Fig. 2 for (a) low, (b) medium, and (c)
+high excitation amplitude; four time snap-shots are depicted
+as (i)-(iv), and the center black line depicts the wavenum-
+ber corresponding to the excitation frequency as given by the
+linear dispersion relation.
+all simulations will be given subsequently.
+Fig. 2 depicts the results for a representative simu-
+lation with a 5-VI configuration (cf. Fig. 1) for low,
+medium, and high forcing amplitude (equivalently low,
+medium, and high energy simulations) corresponding
+A = 0.1, 1, and 10 N, respectively.
+The resulting en-
+ergy measures are computed directly by considering only
+the kinetic energies of the oscillators, which is a reason-
+ably sufficient measure of the total energy distribution
+as elastic systems undergo continuous transfers from ki-
+netic to potential energy. At low amplitude, the acoustics
+are entirely linear as the wave does not create deflections
+greater than the VI clearance (Fig. 2(ai)). The interac-
+tions of the VI mechanisms come about in the medium
+and high amplitude simulations, whereby the energy of
+the propagating wave wave scatters profoundly in the
+space/time domain (Figs. 2 (bi,ci)).
+In the following exposition we provide the results of
+post processing analysis of the measured responses of
+the lattices, with the aim to understand of how the VIs
+scatter the energy of the propagating wave in the fre-
+quency/wavenumber domain. To this end, we utilize a set
+of signal processing procedures which are briefly detailed
+in Appendix A. Figs. 2(aii)-(cii) depict the wavenumber
+distributions across the lattice computed over progres-
+sions of time snap shots for each simulation. Given the
+total collection of simulation data over time and space to
+be the matrix u(x, t), the wavenumber domain at a given
+
+5
+time snap shot, tj, is given as K(κ) = Fx{u(x, t)|t=tj}
+where Fx{ } denotes the Fourier transformation with re-
+spect to the variable x. It is clear from Figs. 2(aii)-(cii)
+that the linear system (corresponding to low excitation
+amplitude) does not affect the wavenumber distribution
+after excitation ends, as expected for a LTI system. In
+contrast, new wave numbers emerge for medium and high
+excitation amplitudes. However, for the case of high en-
+ergy level, the wavenumber generation is not nearly as
+pronounced compared to medium energy level, indicat-
+ing that the wave reflections of Fig. 2(ci) do not generate
+substantial wavenumber components beyond that of the
+incident wave.
+Taking the Fourier transformation across both time
+and space provides the numerically resolved dispersion
+D(κ, ω) = Fx,t{u(x, t)} which is given in Figs. 2(aiii)-
+(ciii).
+Note that Figs. 2(aiii)-(ciii) consider the en-
+tire time record of the simulation from start to finish.
+Fig. 2(aiii) may serve as a reference since no VIs engage
+in the low amplitude simulations, and it is seen that
+only a narrow subset of the dispersion branch is ener-
+getic, corresponding directly to the excitation wavenum-
+ber.
+In the nonlinear regimes, the scattering of the
+energy in the ω-κ domain is much more profound for
+medium energy cases, corroborating the trends estab-
+lished by Figs. 2(i,ii).
+Note that the spectral content
+generated by scattering in Fig. 2(biii) remains bound to
+the underlying linear dispersion relation. Given that the
+VI nonlinearity represents a nonresonant energy scatter-
+ing mechanism, this indicates that the VIs ”redistribute”
+(scatter) wave energy across the dispersion relation of the
+underlying linear lattice rather than modify the disper-
+sion altogether; this acoustical nonlinear scattering effect
+is directly equivalent to the nonresonant scattering mech-
+anisms studied in modal dynamics [70].
+Information regarding the spatial evolution of the gen-
+erated wavenumber components over space and time re-
+quires a space-frequency analysis routine. To this end, we
+employed the continuous wavelet transformation (CWT)
+using the Morelet wavelet in the spatial dimension to
+resolve at each time snap-shot, tj, a 2-D map of the
+wavenumber spectrum with respect to space, X(κ, x) =
+W{u(x, tj)}.
+Fig. 3 depicts the evolution of the spa-
+tial wavenumber distribution tracking X(κ, x) through
+four time snap-shots (t1-t4) for low, medium, and high
+amplitude simulations.
+From this, it is clear that the
+scattering of energy is relatively uniform with respect
+to wavenumber, and that the spectral energy scatters to
+both higher and lower wave numbers (as is also confirmed
+in Fig. 2(ii)). Moreover, the VI-generated wavenumber
+components arise for both the transmitting and reflect-
+ing waves at the VI interface for medium amplitude ex-
+citations, whereas high-energy waves seemingly reflect a
+majority of the incident energy off the VI unit cell at
+the incident wavenumber.
+Lastly, it is apparent from
+Fig. 3(b) that certain wavenumber components propa-
+gate much faster than others and all follow behind the
+incident wavenumber; this is a direct consequence of
+FIG. 4. Propagation of wave energy at different wavenumber
+bands: (a) The kinetic energy versus time at each wavenum-
+ber partition for a mid-energy simulation with sub-panels
+(i)-(vii) plotted to the same color-scale to compare relative
+energies; (b) superimposition of wave propagation at each
+wavenumber partition depicted by contours for (i) low, (ii)
+medium, and (iii) high energy simulation; (c) the optical band
+of the linear lattice plotted with corresponding colors to the
+wavenumber-based energy contours of (b).
+the dispersion relation of the underlying linear system
+(cf. Fig. 1) which is steepest towards the center of the
+optical band and therefore corresponds to larger group
+velocity at the incident wavenumber. Note that this is
+of course not the case when the excitation wavenumber
+is low or high on the band, as the group velocity of the
+incident wave would invariably be smaller for these ex-
+citations. However, the general trends of spectral gener-
+ation with respect to energy are consistent nevertheless
+(see supplemental information).
+The spectral content of Fig. 3 can be mapped-back
+into the spatial-temporal domain by considering a spec-
+tral partitioning scheme similar to that presented in [74].
+The goal is to visualize the propagation of the wave spe-
+cific to different partitions of the optical band, and thus
+confirm that wave propagation at new wavenumbers oc-
+curs due to VI interactions. To achieve this, the instan-
+taneous velocities and positions over various regions of
+the band structure can be resolved by partitioning the
+wavelet space into 12 wavenumber bins and taking the
+inverse wavelet transform of each bin independently. If
+the spatial wavelet-transformed data at a time instant tj
+is denoted as X(κ, x)
+��
+t=tj, and the inverse wavelet trans-
+formation is denoted as W−1, then the dynamics of each
+of the optical band, K1-K12, are computed as the collec-
+
+6
+tion of binned inverse transformations of binned wavelet
+data over time:
+K1(x, t) =
+�
+j
+W−1(X(κ, x))
+��
+t=tj,
+0 ≤ κ ≤ π
+12
+...
+...
+...
+K12(x, t) =
+�
+j
+W−1(X(κ, x))
+��
+t=tj,
+11π
+12 ≤ κ ≤ π.
+(5)
+The kinetic energy can subsequently be computed for
+each spatial-spectral partition, which cannot be achieved
+directly in the frequency domain due to the mass depen-
+dency of the kinetic energy. Summing the energy compo-
+nents of each of the spectral partitions results in negligi-
+ble error (1% or less) compared to the energy computed
+directly from physical coordinates with no numerical in-
+tegral transformations (see supplemental material), thus
+verifying the efficacy of the post-processing technique.
+More importantly, as discussed below, the described nu-
+merical partition of the optical band enables us to study
+in detail the transmission of wave energy at different
+wavenumber bands, and, hence, can shed insight into the
+nonlinear physics of the scattering of the incident wave
+at the VI sites.
+Fig. 4 depicts the results of the wavenumber parti-
+tioning scheme. The propagation of energy across each
+wavenumber partition are given by subplots 4(ai)-a(vii)
+and plotted to the same color scale in order to com-
+pare the relative energies of each wavenumber partition.
+The wave initiates in K7 and K8 as these posses energy
+from the onset of propagation while all other wavenumber
+partitions are dormant during the start of propagation.
+However, after the VIs are engaged midway through the
+lattice, energy begins to propagation through all parti-
+tions, and this is clear indication that the VI nonlinear-
+ity in fact generates wave propagation at wavenumbers
+not native to the excitation profile. To demonstrate the
+dependency on energy, Fig. 4(b) shows the wave propa-
+gation through each wavenumber band superimposed by
+contours for low, medium, and high profile wavenumber
+from which it is apparent again that wavenumber genera-
+tion is far more potent at medium amplitude simulations
+than for high ones. Fig. 4(c) provides a colored depiction
+of the optical band to make the contours of Fig. 4(b) more
+obvious with respect to which wavenumber components
+are generated; the of group velocities in Fig. 4(c) corre-
+sponds directly to the variable wave speeds of Fig. 4(b),
+and this can be used to interpret the variation in spatial-
+spectral propagation of Fig. 3 as well.
+C.
+Quantifying Wavenumber Spectrum Disorder
+Section II B established that (i) the VIs generate prop-
+agating waves at new wavenumbers as they interact with
+the incident wave, and (ii) that this phenomenon is de-
+pendent on amplitude. To this point, the results have
+FIG. 5.
+Mean spectral entropy in the lattice with VIs for
+system configurations ranging between 1 VI to 20 VI (see
+Fig. 1) over an array of excitation amplitudes logarithimcally
+spaced from 0.1 to 20: Top and bottom plots are for the same
+data with the bottom plots depicting the log-log scaling; a
+fitted power law is denoted as a thick black line, and the
+adjusted R-squared value is listed for each configuration in
+the bottom plots.
+been presented in a largely qualitative manner with an
+emphasis on graphical interpretations (cf. Figs. 2,3,4).
+We now aim to quantify the wavenumber scattering in-
+duced by the VIs for wave transmission over the entire
+domain of the lattice, based on an ensemble of simula-
+tions, in order to establish the dependence of VI induced
+wavenumber scattering on input amplitude.
+To this end, we make use of information theory by
+considering the spectral entropy of the nonlinear acous-
+tics in the wavenumber domain. Spectral entropy is the
+extension of classical Shannon entropy to the frequency
+domain [75] and is a standard metric for quantifying sig-
+nal complexity.
+We consider the wavenumber entropy
+generated over space at a given time snap shot as
+H(x) = −
+�
+κ
+P(x, κ) log2 P(x, κ),
+(6)
+where P(x, κ)
+=
+S(x, κ)
+��
+ξ S(x, ξ) is the space-
+dependent probability distribution over wavenumber
+computed with the space-frequency power spectrogram
+S(x, κ).
+By computing P(x, κ) over a progression of
+time snapshots, tj, for each simulation, a matrix of
+entropy-versus-time, H(x, t), captures the time-evolution
+of wavenumber entropy as the wave propagates through
+the lattice.
+We compute a statistical summary of the
+wavenumber entropy by considering the elements of
+H(x, t) for time intervals after the incident wave has al-
+ready reached the first VI unit cell at t = ˆt. Fig. 5 depicts
+the normalized average entropy quantity with respect to
+forcing amplitude for all configurations depicted in Fig. 1.
+Normalization was performed so that the minimum and
+maximum entropy for each VI configuration range be-
+tween 0.01 and 1. To this effect, we are capturing the
+
+7
+relative scattering of wavenumbers as compared to an op-
+timal excitation amplitude (specific to our selected con-
+figuration). At the lowest forcing excitation level (with
+no VI engagement) the wave propagation remains lin-
+ear, and so the entropy remains nearly zero as the only
+variation in the wavenumber comes from the intrinsic dis-
+persive characteristics of the lattice. However, once the
+VIs are engaged at medium and high excitation levels,
+the entropy rapids rises and reaches a maximum before
+rapidly falling again with respect to forcing amplitude.
+The log-log plots of Fig. 5 reveal that after the maximum
+entropy is reached, the remainder of the data fits remark-
+ably well with a power law with adjusted R-squared co-
+efficients above 0.95 being recovered for the majority of
+configurations studied. Error bars in Fig. 5 measure the
+standard deviation of entropy across the spatial extent of
+the lattice. This can be interpreted as a measure of how
+uniform the wavenumber complexity is. Hence, the larger
+error bounds at high excitation amplitudes indicate that
+novel wavenumber components are localized rather than
+distributed (or propagated) throughout the spatial ex-
+tent of the lattice, and this is in direct agreement with
+the qualitative results of Figs. 2, 3, and 4. Note that the
+excitation wavenumber is κ = 5π/9 for all results shown
+in Fig. 5; additional results given in the supplemental
+material confirms that the same trends hold across all
+incident wavenumbers.
+III.
+INTER-BAND TARGETED ENERGY
+TRANSFERS (IBTET)
+With section II establishing that the VI nonlineari-
+ties can scatter energy about the optical band of a di-
+atomic lattice, a natural next question is to what effect
+VI mechanisms can induce targeted energy across dif-
+ferent bands.
+This can be considered as an acoustics-
+equivalent to the IMTET nonlinear mechanism estab-
+lished in dynamics [68]. Hence, the aim of this section is
+to achieve inter-band targeted energy transfers (IBTET)
+by irreversibly transferring energy from a lower optical
+band to a higher band. Moreover, we aim to demonstrate
+that this phenomenon is achievable for multiple classes of
+VI contact laws, and introduce a bilinear version of the
+VI law considered previously, to be studied alongside the
+Hertzian model of Section II. This is considered in order
+to demonstrate that the subsequent IBTET results are
+reproducible for different classes of contact nonlinearity
+and are not particular to the Hertzian contact law utilized
+in section II, hence opening a broader design space to re-
+alize the phenomenon in practice.
+To achieve IBTET
+requires a system with more than 2 DoF per unit cell,
+since the number of optical bands amenable to out-of-
+phase motion, and thus with the ability to interact with
+the VI, is dictated by Noptical = NDoF −D where NDoF is
+the degrees of freedom in the unit cell and D is the unit
+cell dimension. Hence, to maintain the simplicity of 1D,
+we proceed with a 4-DoF model of the unit cell, offering
+(a)
+(b)
+UNIT CELL
+FIG. 6. Increasing the bands of the lattice: (a) Schematic of
+the unit cell, and (b) the corresponding dispersion diagram
+for parameters λ = 0.1 and η = 0.5.
+two additional bands to transfer energy towards.
+A.
+The 4-band Lattice
+The 4-band model emulates closely the resonator
+model of Fig. 1. The main difference is that masses have
+been added in-series in between resonators as shown in
+Fig. 6(a). The equations of motion for a unit cell of the
+infinite 4-band phononic lattice read,
+m1¨uk
+1 + k4
+�
+uk
+1 − ui−1
+4
+�
++ k1
+�
+uk
+1 − uk
+2
+�
+= 0
+m2¨uk
+2 + k2
+�
+uk
+2 − uk
+1
+�
++ k3
+�
+uk
+2 − uk
+3
+�
++ k4
+�
+uk
+2 − uk
+4
+�
++ fNL(wk) = 0
+m3¨uk
+3 + k3(uk
+3 − uk
+2) − fNL(wk) = 0
+m4¨uk
+4 + k1
+�
+uk
+4 − ui+1
+1
+�
++ k4
+�
+uk
+4 − uk
+2
+�
+= 0
+(7)
+which produces a 4-band dispersion relation upon appli-
+cation of the Bloch theorem. To maximize the potential
+for IBTET, the parameters of system (7) should be se-
+lected to satisfy the following criteria:
+• The displacements of the host-mass and resonator
+of the VI oscillator (u2 and u3) should be out-of-
+phase on the second band so that strong engage-
+ment of the VI nonlinearity can occur beneath the
+2nd and 3rd optical bands (since VIs transfer en-
+ergy from low-to-high frequencies [70]).
+• The quantity | ˆw| = |ˆu3(κ) − ˆu2(κ)| describing
+the resonator deflection across the second Bloch-
+eigenmode should be maximized over κ on the sec-
+ond band.
+• The group velocity corresponding to the second
+band should be as high as possible in order to mini-
+mize the dispersive effects originating from the lin-
+ear band structure.
+• The group velocities of the third and fourth bands
+should be maximized so as to maximize the cor-
+responding band slopes and equivalently broaden
+the bandwidth that is amenable for TET from the
+second band.
+
+8
+FIG. 7. IBTET in the 4-band lattice with 5 VI sites: (a) shows the evolution of the propagating wave energy; (b-e) propagation
+of the wave energy corresponding to each band of the lattice based on the numerically recovered dispersion of the full simulation;
+(f,g) dispersion of the input and output segments (labeled in (a)) demonstrating the targeted energy transfer to the higher
+bands; (h,i) Fourier spectra corresponding to the velocity of the four unit cell DoFs selected before (5-th unit cell) and after
+(150-th unit cell) VI engagement, with the four band-pass regions depicted with shading and insets depicting the corresponding
+velocity time histories.
+System (7) is parameterized by η and λ which relate
+the mass and stiffness of the resonator cell to the nominal
+parameters of m1 = m4 = m = 0.005 kg and k1 = k4 =
+k = 2 × 104 N/m by m2 = m(1 − η), m3 = mη, and k3 =
+kλ while we fix k2 = 104 N/m. With these variables, the
+desired dispersion characteristics can be readily achieved
+by considering a cost-function of the form
+max
+η,λ
+� 4
+�
+k=2
+|vk
+g|
+�
+w,
+s.t. ˜u2(κ)˜u3(κ) < 0 ∀κ.
+(8)
+We confine this search for 0.1 < λ < 1 and 0.1 < η < 1.
+With this constraint, minimizing the cost function over
+(λ, η) is trivial and returns λ = 0.1 and η = 0.5. The
+resulting band structure is shown in Fig. 6(b).
+To simulate the system, a finite lattice of 300 unit cells
+(1200 DoF) was constructed, which is one half of the
+total DoFs of the resonator chain studied in section II.
+Accordingly, we consider only a 5-VI lattice configura-
+tion (as depicted in Fig. 1(d)) herein and refer the reader
+to supplemental material for the results of a 1-VI lattice
+configuration. Simulations were performed similarly to
+section II with excitation provided by a windowed tone
+burst (Eq (3)). An input signal of 30 periods was con-
+sidered, and the excitation frequency is selected based on
+the maximum group velocity of the optical band. Simu-
+lations were performed for 50 selections of the excitation
+amplitude between 1 and 104 N.
+We employ the same Hertzian contact law described
+by Eq (4) for n = 3/2, and also a bilinear contact law
+which takes the same form as Eq (4) but for n = 1. This
+is performed to ensure that the subsequent results are
+not particular to nonlinear Hertzian contact laws but are
+rather a product of the contact nonlinearity. For the 4-
+band system considered, the contact stiffness parameters
+(kc) were computed based on E = 100 MPa, ν = 0.3,
+and RVI = 0.005 m, and the clearances are now varied
+between 10−2.65 and 10−2.75 m.
+B.
+Low-to-high band targeted energy transfer
+Fig. 7 depicts an example of a wave propagating
+through the 4-band system with five Herzian VIs en-
+gaged.
+Energy clearly cascades from the main wave
+packet as it propagates through the lattice (Fig. 7(a)),
+similar to the diatomic chain (Fig. 2). Computing the
+numerical dispersion at the beginning and end of the sim-
+ulation clearly shows that energy in fact transfers from
+the lowest optical band to the higher two optical bands
+(Figs. 7(f,g)). This is further confirmed by Figs. 7(h,i)
+which shows the difference in the temporal frequency
+of the wave at the start versus end of the lattice and
+
+9
+FIG. 8. IBTET depicted in terms of the dispersion of the wave
+in the frequency/wavenumber domain for the 4-band lattice
+with 5 VI sites over the entire duration of the simulation for
+(a) Hertzian and (b) bilinear VI laws, and for (i) low, (ii)
+medium, and (iii) high excitation amplitudes.
+hence the low-to-high frequency targeted transfer of en-
+ergy from the second band to the higher bands.
+Energy transfer between bands can be quantified by
+first converting the numerically measured data into the
+ω-κ domain with the 2-D Fourier transformation. There-
+after, the 2-D spectrum is partitioned band-by-band and
+also into band-gap regions. For each partition, the re-
+mainder of the spectrum is zero-padded before the inverse
+Fourier Transformation returns the spectral content into
+the spatio-temporal domain for that specific partition.
+This results in the propagation depicted in Figs. 7(b-e)
+where it can be seen that the content of the upper bands
+indeed corresponds to propagating waves generated by
+the VIs, and thereafter kinetic energy calculations over
+each band can be conveniently performed.
+Fig. 8 depicts the numerical dispersion of both the
+Hertzian and bilinear systems for low, medium, and high
+excitation amplitudes, which shows that the most pro-
+found energy transfer occurs in the medium amplitude
+range, much like what was seen in section II. Note that
+these low, medium, and high excitation amplitudes now
+refer to order 1, order 10, and order 100 N. To verify and
+quantify the efficacy of the VIs to induce TET from low-
+to-high bands (i.e., to induce IBTET) with respect to
+excitation amplitude, the energy stored within the up-
+per two optical bands is recovered and normalized per
+the total system energy. This normalized energy is time-
+averaged taking into account only the time window after
+the propagating wavefront encounters the first VI site in
+the lattice.
+FIG. 9. The portion of input energy transferred to the upper
+two optical bands versus forcing amplitude of the incident
+wave for (a) Hertzian VIs and (b) bilinear VIs in (i) depicting
+linear-linear and (ii) log-log scales.
+Fig. 9 depicts the results of the IBTET analysis over
+the ranges of forcing amplitudes considered for both
+Hertzian and bilinear VI laws. The log-log plots depict
+a very similar trend to what was observed in section II:
+a sudden spike in energy transfer once the amplitude is
+sufficient enough to engage the VI, and a sudden decline
+in energy transfer as the excitation amplitudes rise there-
+after. The portion of the energy transferred to the higher
+bands continues to fall until it reaches a minimum defined
+by the relative energy obtained by the higher bands for a
+completely linear system. This is on the order of 0.01 %
+of the total system energy, and is of course explainable
+by the fact that the windowed tone burst used to excite
+the system assumes a Gaussian distribution in the fre-
+quency domain which invariably provides trace amounts
+of energy across the entirety of the spectrum due to the
+Fourier uncertainty principle.
+Interestingly, the same trends in IBTET are observed
+for both Hertzian and bilinear contacts, indicating that
+the nature of the contact law does not play a critical
+role in the energy transfer, but rather the discontinu-
+ous potential is the driving mechanism for the energy
+exchanges. This is further verified in the linearly-scaled
+plots of Figs. 9(aii,bii) which show that the maximum
+energy transferred to the higher optical bands is roughly
+0.3-0.35 (30-35%) for both the Hertzian and bilinear VIs.
+Not only does this demonstrate that a substantial portion
+of energy may be irreversibly transferred to higher bands,
+but that this is achievable for a variety of VI designs,
+opening broader designs avenues for practical acoustic
+
+4523232222810
+FIG. 10. A 2-DoF model emulating a VI resonator cell.
+metamaterials that could exhibit IBTET.
+IV.
+PHYSICAL INTERPRETATION OF IBTET
+MECHANISM
+We now seek to connect the trends established in Sec-
+tions II and III to physics-informed arguments in order
+to shed physical insight into IBTET in a consistent and
+comprehensive way. We do so by considering a reduced
+order model (ROM) of a VI-oscillator to emulate the VI
+unit cells embedded in the finite lattices, and then in-
+terpret IBTET by studying the nonlinear normal modes
+(NNMs) of the ROM. NNMs have proven a useful tool
+for interpreting the responses of nonlinear dynamical sys-
+tems and their passive tunability with respect to energy
+through either analytical or computational tools [76–79].
+The uses and interpretations of NNMs are quite exten-
+sive, however a direct and intelligible way of interpret-
+ing the evolution of the system’s dynamics with respect
+to energy is with the frequency energy plot (FEP) of a
+given dynamical system and its bifurcating branches [76].
+Such methodology has been employed already for under-
+standing the dynamical evolution of VI systems of various
+forms [71, 80, 81].
+A.
+Reduced Order Model (ROM)
+We consider a 2-DoF ROM that is designed to emulate
+the individual VI-resonators embedded within the 4-band
+lattice of section III. Fig. 10 provides a schematic of the
+ROM whereby the parameters ¯k1 = k = 2 × 104 N/m,
+¯k2 = 2 × 103 N/m, and ¯m2 = ¯m2 = 0.0025 kg, which
+parameterize the set of equations
+¯m1¨¯u1 + ¯k1¯u1 + k2(¯u1 − ¯u2) + fNL( ¯w) = 0,
+¯m2¨¯u2 + ¯k2(¯u2 − ¯u1) − fNL( ¯w) = 0.
+(9)
+where an overbar denotes that the variable is associated
+with the ROM and not the full phononic lattice. The
+nonlinear force fNL( ¯w) in Eq (9) is taken with respect to
+¯w = ¯u1 − ¯u2, where VI nonlinearity is considered as both
+Hertzian and bilinear form with a contact stiffness and
+clearance of 10−2.75 m.
+A key difference to note is that the ROM has fixed
+boundaries, whereas the resonator embedded within VI
+unit cells of the full phononic lattice does not. However,
+we assume that the stiffness between masses in the lat-
+tice is distributed between the two mass elements, and
+thus the total stiffness of the ROM host mass with re-
+spect to its equilibrium position can be approximated by
+considering that fixed boundaries with one-half the total
+stiffness of the flexible boundaries of the full phononic
+lattice.
+Moreover, the most critical component of the
+ROM is the internal stiffness and nonlinear VI compo-
+nent, which matches identically to the VI cells consid-
+ered in Section III. Hence, the ROM provides reasonable
+resemblance to the VI cells in the full lattice system al-
+lowing it to capture the trends of the full system with
+surprisingly good accuracy, as we will show.
+B.
+Nonlinear Normal Modes as a Measure of
+Nonlinearity
+The energy dependencies of Figs. 5 and 9 make an
+NNM approach a natural avenue since continuation re-
+turns an overview of the dynamics across energy scales.
+To this end, we compute the NNMs of the ROM by em-
+ploying a continuation scheme described in [79] with mi-
+nor modifications listed (see Appendix B). We provide a
+grossly condensed description herein and refer the reader
+to [79] for full algorithmic details. The state form of sys-
+tem (9) is ˙z = g(z) where g(z) is a nonlinear function
+of the state variables. A periodic orbit (or NNM) will
+satisfy the two-point boundary value problem defined by
+the shooting function, H(zp0, T) = z(zp0, T) − zp0 = 0.
+Newton’s method can be used to recover periodic solu-
+tions at low energy in the shooting stage. We define the
+phase condition such that the two DoFs of the ROM
+have zero initial velocities. After shooting is completed,
+a pseudo-arclength method is used to trace out the NNM
+branch in the 2n + 1 dimensional parameter space. In
+brief, this works by computing predictor steps using the
+tangent vector at the most recently converged solution,
+and then making corrector steps in an orthogonal direc-
+tion to the tangent until convergence is achieved. This is
+a critical step for resolving the NNMs of the VI system
+since the NNM branches may have turning points that
+the standard Newton-Raphson algorithm cannot solve.
+The result of numerical continuation is a frequency
+energy plot (FEP) which describes the evolution of the
+NNM branch for 1:1 resonance (the so called “backbone”
+branches) in the frequency-energy space. Fig. 11 depicts
+the FEPs computed for system described by equation (9)
+for both Hertzian and bilinear contact laws. It is inter-
+esting to emphasize that the degree (strength) of non-
+linearity of the ROM can be qualitatively interpreted by
+the slope of a given NNM branch [71]. The steeper the
+slope is of the branch is, the more sensitive the frequency-
+amplitude dependency of the NNM becomes, and the
+more intense the nonlinearity in the ROM when it re-
+sponds on that NNM is.
+The FEP results reveal similar trends for both Hertzian
+
+11
+FIG. 11. The FEPs of the ROMs with (a) Hertzian and (b)
+bilinear nonlinearity with insets zooming in on the transi-
+tion from region I to II with instability denoted by orange
+for regions with Floquet multipliers |α| ≫ 1; (c,d) slopes of
+the FEPs of of (a,b) with respect to energy; (e) and (f) cor-
+responding phase trajectories of the NNMs for (a) and (b),
+respectively, for regions I, II, III, and IV of the FEPs.
+and bilinear VI ROMS, possessing four dynamical region
+labeled (I)-(IV) in Fig. 11. The corresponding phase tra-
+jectories of the periodic orbits in each region are given
+in Fig. 11(e) and 11(f) for Hertzian and Bilinaer mod-
+els, respectively. In the low energy region (I), the VIs
+do not engage, and the dynamics are completely linear;
+this is confirmed by zero slope of the FEP. In region
+(II), there is a grazing of the VI contacts, causing a sud-
+den change in the dynamics and a rapid increase of FEP
+slope. In fact, the corresponding NNM branch folds back
+on itself and goes backwards in energy before re-directing
+again towards higher energies, with this effect being more
+prevalent in the bilinear model (the Hertzian nonlinear-
+ity being less prominent in the small deflection amplitude
+limit). This in turn yields a small neighborhood of the
+NNM branch where the FEP slope is theoretically infi-
+nite, and the subplots of Figs. 11(c,d) confirm that this
+is where to maximum is reached. The phase trajectories
+indicate that region II represents a transition where the
+dynamics are most sensitive to nonlinear effects. Despite
+the apparent smoothness of Figs. 11(eII,fII) the volatile
+VI-grazing dynamics in region II are unstable, and hence,
+not physically realizable. Computation of NNMs in this
+regions requires Newton predictions on a similar order of
+machine tolerance and results in strongly unstable NNMs
+as depicted in Fig. 11 for portions of the NNM branch
+with Floquet multiplier, α, far exceeding 1.
+After the grazing VI region in region II is surpassed
+with increasing energy, the FEP gradually increases in
+frequency towards region III. Region III is character-
+ized by strong VI oscillations which is apparent by the
+box-like phase trajectories indicating non-smooth tem-
+poral dynamics. In this region, the linear dynamics of
+ˆk1 are negligible and the VI dynamics dominant. Note
+that it is in region III that the slopes of the FEPs de-
+crease in a power-law like fashion as the ROM asymp-
+totically reaches the limiting region IV. Region IV mani-
+fests smooth dynamics characterized by in-phase dynam-
+ics predominantly dictated the contact stiffness. In this
+region, the clearance is negligible and the VI contacts be-
+have as an extremely stiff elastic spring. Hence, the dy-
+namics of the ROM with Hertzian contacts approaches a
+smoothly nonlinear system with a 3/2 nonlinear coupling,
+whereas the dynamics of the bilinear ROM approaches a
+linear system at high energy, as is confirmed by the phase
+portraits of Figs. 11(eIV,fIV). Moreover, for the bilinear
+system, the FEP clearly levels off as the high-energy (al-
+most) linear limiting behavior is reached.
+C.
+Relating the Dynamics of the ROM to the
+Acoustics of the Lattice
+The evolution of the FEP slope with respect to energy
+of the ROM (Figs. 11(b,c)) posses a remarkable similar-
+ity to the observed trends of nonlinear IBTET in the
+full phononic lattice (Fig. 9). The two measures can be
+related to one another by replotting the energy trans-
+fers of Fig. 9 with respect to system energy (to match
+the energy-dependent nature of the FEP) and superim-
+posing the FEP slopes to compare similarities in their
+evolution with energy. To do this requires a normaliza-
+tion, as the maximum and minimum values of the FEP
+slope can be arbitrarily large or small, whereas the rela-
+tive energy of the upper optical bands is lower-bounded
+by the amount provided by the excitation source (from
+the Fourier uncertainty principal), and upper-bounded
+by unity (since the energy in the upper bands cannot
+exceed the total energy of the system). Moreover, the
+wave propagation in the 1200 DoF phononic lattice car-
+ries the energy of 30 cycles of the windowed excitation,
+whereas the FEP energy is parameterized by the periodic
+orbits of the 2 DoF ROM. Thus, the energy of the finite
+lattice must be normalized in order to be commensurate
+with the energy of the ROM used to generated the FEP.
+These normalizations are performed as follows. The FEP
+
+12
+FIG. 12. The relative inter-band energy transfer, with the
+normalized slope from the ROM-FEP superimposed for (a)
+Hertzian and (b) bilinear contact models; the dashed lines
+depict the normalized FEP slopes, the gray lines depict the
+normalized FEP slopes lower-bounded by the initial (linear)
+energy of the higher bands, and green lines depict a power
+law fit to red dots, with the adjusted R-squared value shown
+with the inset.
+slope is divided by a scalar as to quantitatively align with
+the relative energy transfer in quantity so that a direct
+comparison can be made with respect to decay rate ver-
+sus energy. A scalar quantity defined by the low-bound
+of IBTET (dashed lines of Fig. 9) is then added to the
+FEP slope account for the lower threshold of the energy
+transfer in the VI lattice. The energy of the finite lattice
+is normalized so that the initiation energy, that is, the
+energy required to engage the first VI site encountered
+by the propagating wavefront, aligns with the transition
+between regions I and II of the FEP. These normaliza-
+tions preserve the slopes of both quantities since scalar
+multiplication results only in translations in log scaling.
+Hence, the previous measures can be directly compared
+with respect to their decrease in value with respect to
+increasing normalized energy.
+Fig. 12 displays the described superposition where a
+remarkable agreement is found between the trends in the
+slope of the FEP of the ROM and the energy transfer be-
+tween bands in the lattice. Hence, the underlying FEP
+of the ROM, along with the evolution of the dynamical
+regimes of Fig. 11, clearly have a direct implication of
+the IBTET in the lattice. Moreover, by fitting a slope
+to the measured energy transfer versus normalized sys-
+tem energy for data points falling in region III, a near-
+perfect power law is recovered as indicated by the ad-
+justed R-squared values close to 1 (see Fig. 12). Finally,
+these results are in agreement with the trends observed
+for wavenumber spreading within the optical band of the
+2-band system considered in section II. Hence, the nu-
+merical results presented for the finite lattices can be
+understood based in terms of the underlying nonlinear
+dynamics of the ROM based on the single VI unit cell as
+it transitions between various dynamical regimes with re-
+spect to energy. With this, a predictive tool is presented
+to assess the capacity for IBTET in full phononic sys-
+tems based on the simplified VI ROMs which, being of
+low-dimensionality, are much more amenable to analysis
+compared to the extended nonlinear lattices considered
+herein.
+V.
+CONCLUSIONS
+In this work, we have investigated the effect of local
+VI nonlinearities on the propagation of traveling waves
+in 1-D phononic lattices. Specifically, first a di-atomic
+2-band lattice was numerically studied over a wide range
+of forcing amplitudes and embedded VI configurations
+(section II). It was demonstrated that wavenumber scat-
+tering in the optical band of this lattice is most pro-
+found for moderate excitation amplitudes, and decreases
+in effectiveness as the energy rises (Fig. 2).
+This was
+quantified by considering the spatial-spectral entropy (or
+wavenumber entropy), for various systems which all fol-
+lowed very closely to power-law decays with respect to
+excitation amplitude after the peak value was reached
+(Fig. 5). Attention then turned to inter-band targeted
+energy transfer (IBTET) in a 4-band system which was
+parameterized in order to provide dispersion curves re-
+ceptive to such energy transfers (Section III). Simula-
+tions were carried out over a range of excitation ampli-
+tudes with both Hertzian and bilinear contact laws. Nu-
+merical post-processing reconstructed the energy of each
+band, and it was shown that IBTET is indeed possible.
+Moreover, this phenomenon was proven effective for both
+Hertzian and bilinear VIs, and the trends in IBTET with
+respect to excitation amplitude followed closely to those
+observed for wavenumber scattering in the 2-band lattice
+(Fig. 9).
+In an attempt to shed some physical insight into the
+effect of the VIs on the acoustics of the lattice, a low-
+dimensional ROM was constructed based on the unit
+VI cell. The underlying FEP of the 2 DoF ROM was
+computed for the NNM family of 1:1 resonance branches
+which revealed four dynamic regimes that the ROM as-
+sumes with respect to energy. Namely, a linear low en-
+ergy region, a grazing region initiated when the VI non-
+linearity first enters the dynamics, a full VI-oscillator
+with nonsmooth temporal dynamics, and an effectively
+linear or smoothly nonlinear high-energy regime, depend-
+ing on the contact law (Hertzian or bilinear). This, in
+turn, produced a frequency-energy slope that directly
+scales to the trends of IBTET in the lattice with respect
+to system energy, providing the physical interpretation of
+the spectral scattering of sections II and III. Moreover,
+the FEP presents a means for accurately predicting en-
+
+13
+ergy transfer capacity of the full phononic lattice based
+on the low-dimensional ROM.
+Although this work focused primarily on fundamental
+understanding of the physics at play, the implications and
+potential for future developments are rather extensive.
+The low-to-high energy transfers directly correspond to
+a reduction in magnitude, since the energy must be pre-
+served in the frequency transfer. Moreover, the evolution
+of the VI dynamics with respect to energy corresponds
+to an effective filter that can greatly alter transmissibil-
+ity of incident waves (cf. Fig. 3). These attributes alone
+make VI-based methods attractive for wave transmission
+tuning (or tailoring) with respect to amplitude. More-
+over, while we have targeted low-to-high energy transfers
+between bands, future works could explore the potential
+for targeting specific bands and specific sub-regions of
+bands of phononic lattices by optimizing the distribution
+and parameters of local VIs in lattices through methods
+such as genetic programming or machine learning.
+ACKNOWLEDGMENTS
+This work was supported in part by the National Sci-
+ence Foundation Graduate Research Fellowship Program
+under Grant No. DGE – 1746047. Any opinions, find-
+ings, and conclusions or recommendations expressed in
+this material are those of the authors and do not neces-
+sarily reflect the views of the National Science Founda-
+tion.
+A.
+DETAILS ON SIGNAL PROCESSING
+PROCEDURES
+1.
+Continuous Wavelet Transformation (CWT)
+In this section, we provide a brief discussion of the
+wavelet transformation algorithm employed in this work
+in order to clarify the mathematical details pertinent
+for performing the wavelet-based wavenumber partition
+analysis of section II (cf. Fig. 4). A similar discourse
+may be found in [74]. The CWT is traditionally used as
+a time-frequency analysis tool by transforming the signal
+from the time domain to the time-frequency domain. To
+the same effect, one can consider the space-wavenumber
+domain. For 1D systems the standard definition of the
+CWT with respect to the spatial variable x is,
+X(x, κ) =
+� κ
+κc
+� ∞
+−∞
+u(ξ)ψ∗
+�ξ − x
+κc
+�
+dξ
+(10)
+where ψ∗(ξ) is the complex conjugate of the mother
+wavelet function and κc the center frequency,
+κc =
+�� ∞
+0
+κ2|Ψ(κ)|2dκ
+� ∞
+0
+Ψ(κ)|2dκ
+�1/2
+.
+(11)
+FIG. 13. The reconstructed kinetic energy and correspond-
+ing reconstruction error for the described wavelet partition
+scheme; red dashed line indicates 1 percent error.
+We consider the Morelet wavelet for all transformations
+in this work:
+ψ(x) =
+1
+π1/4
+�
+eiκcx − e−κ2
+c/2�
+e−x2/2.
+(12)
+For the scale and quantities of datasets considered in this
+work, computational efficiency is a requirement. To this
+end, the Fast Fourier Transform is employed to speed
+up wavelet computations. Taking Ψ(κ) as the analytical
+Fourier Transform of the mother wavelet,
+Ψ(κ) = e−(κ−κc)2/2,
+(13)
+and ˜x(κ) the FFT of the signal, the wavelet transforma-
+tion can be written equivalently as:
+X(κ, x) =
+�κc
+κ
+� ∞
+−∞
+˜x(η)Ψ∗(ηκ/κc)eiηxdη.
+(14)
+Each wavelet transformation can be partitioned over
+space and wavenumber. The spectral partitions are de-
+fined over 12 regions spanning between κ = 0 and κ = π
+to account for 12 different wavelet-domain representa-
+tions of the spatial signal at each time instant. The k-th
+wavenumber partition is defined as:
+Xk(κ, x) = X(κ, x)hk(κ),
+hk(κ) = H
+�
+κ − (k − 1)π
+12
+�
+− H
+�
+κ − kπ
+12
+�
+.
+(15)
+The inverse wavelet transformation can be applied
+at each time snap shot to each wavenumber partition,
+uk(x) = W−1 {Xk(κ, x)}, which is computed as:
+uk(x) =
+√κ
+κ3/2
+c
+C
+� ∞
+0
+� ∞
+−∞
+ˆXk(κ, ξ)Ψ
+�ξκ
+κc
+�
+dξdκ.
+(16)
+where ˆXk(κ, ξ) is the Fourier transformation of Xk(κ, x)
+with respect to x. Fig. 13 depicts the reconstructed ki-
+netic energy of the lattice, KErec, as well as the directly
+computed (exact) kinetic energy from the numerical sim-
+ulations KEphys, with the error between the two quanti-
+ties computed by:
+e(t) = ||KErec(t) − KEphys(T)||
+||KEphys(t)||
+.
+(17)
+
+14
+FIG. 14. Contours of the instantaneous wavenumber entropy
+across the time-entropy domain for low, medium, and high
+amplitude simulations(top), and the summary contours of the
+instantaneous entropy H(t) (bottom).
+2.
+Spectral Entropy
+Here, we provide more details pertaining to the spec-
+tral entropy plots displayed in Fig. 5.
+Fig. 14 depicts
+the distribution of entropy using Eq (6) to recover H(x)
+for each t. The resulting matrix H(x, t) is plotted as an
+image for low, medium, and high excitation amplitudes.
+The distribution of high-entropy regions is clearly seen in
+the medium and high excitation amplitude simulations
+as the VIs engage the incoming wave. Superimposed on
+each image is the instantaneous spectral entropy, which
+summarizes H(x, t) over space to render time-dependent
+measures H(t).
+A data set storing H(t) for each excitation ampli-
+tude in the simulation ensemble can then be generated
+and plotted in the form of an image to study how the
+wavenumber entropy varies in time with respect to the
+forcing amplitude for a given lattice configuration. This
+is depicted in the bottom plot of Fig. 14. In the low-
+amplitude region with no VI engagement, no entropy is
+generated after excitation (as expected).
+For medium
+amplitudes, regions of sustained high wavenumber en-
+tropy are realized after the VIs engage the incident wave.
+In contrast, only localized patches of high entropy are
+seen for high-amplitude simulations, indicating that the
+VIs do not affect the global wavenumber of the lattice
+after the incident wave passes through (or reflects off of)
+the unit cells with embedded VIs.
+FIG. 15. Energy Reconstruction of band-partitioning decom-
+position.
+3.
+Computing energy on each band
+The computation of wave energy over each band in
+section III is performed as follows. The data matrix for a
+given simulation is mapped to the Fourier domain using
+the 2D FFT algorithm D(κ, ω) = Fx,t{u(x, t)}. Next,
+frequency filters are constructed as follows,
+Gk(κ, ω) =
+�
+1
+ω ∈ Bk, −π ≤ κ ≤ π
+0
+otherwise
+(18)
+were the first four ranges of frequencies Bk are defined
+over the temporal frequency limits of the four pass-bands
+(PB),
+B1 = min(PB1) ≤ ω ≤ max(PB1)
+B2 = min(PB2) ≤ ω ≤ max(PB2)
+B3 = min(PB3) ≤ ω ≤ max(PB3)
+B4 = min(PB4) ≤ ω ≤ max(PB4)
+(19)
+A remaining two filter banks are constructed for the band
+gap between the acoustic band and first optical band
+(BG1), and of for the band gap between the upper two
+optical bands (BG2),
+B5 = min(BG1) ≤ ω ≤ max(BG1)
+B6 = min(BG2) ≤ ω ≤ max(BG2).
+(20)
+The spatial-temporal dynamics corresponding to each
+pass band and band gap regions are then given as,
+uk(x, t) = F−x,−t{Gk(κ, ω) · D(κ, ω)}
+where F−x,−t{ } indicates the 2D inverse FFT with re-
+spect to x and t. The rigid boundaries of the filters in
+Fourier space inevitably results in minute numerical ar-
+tifacts in the inverse transformation for each partition
+
+15
+taking the form of ripples along the space-time bound-
+aries. However, the reconstruction of energies computed
+by summing the energy over each band matched nearly
+identically to the energies computed for the direct nu-
+merical simulations, and hence these numerical artifacts
+are negligible.
+B.
+NONLINEAR NORMAL MODE
+COMPUTATIONS
+The recipe for NNM calculations follows very closely
+to the procedure outlined in [79].
+For all FEP calcu-
+lations, the shooting method used a prescribed initial
+step size of 1−5 and a tolerance of ε = 1 × 10−6. For
+low energy orbits, Newmark integration was employed
+with 2000 steps per period, and Jacobian calculations of
+predictor-corrector steps were computed using the sensi-
+tivity analysis in [79]. In region II, the unstable dynamics
+proved to be challenging for the computation of the corre-
+sponding NNM branch. Hence, sufficiently small predic-
+tor steps were required for convergence, with the residual
+reduction being varied from 10−12 to 10−10. Sensitivity
+analysis was employed again to compute Jacobian terms
+in region II.
+Once the dynamics of the NNMs stabilized to that of
+a definitive VI oscillator in region III, and moreover to
+smoothly stable NNMs in region IV, the finite difference
+method sufficiently approximated Jacobian terms allow-
+ing for the implementation of fast and accurate Runge-
+Kutta based methods such as ODE78. The nonsmooth
+nature of dynamics in region III would require still a
+great number of Newmark iterations to achieve the same
+accuracy as the ODE78 routine, and therefore the transi-
+tion was made to a finite-difference Jacobian calculation
+scheme based on ODE78 for energies beyond region II to
+increase computational speed and reduce the number of
+steps required to resolve the high-energy regions of the
+FEP branch.
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+64-65, 266 (2015).
+
+1
+Supplemental Materials: Wavenumber Scattering and Inter-band Targeted Energy
+Transfer in Phononic Lattices with Local Vibro-Impact Nonlinearities
+Joshua R. Tempelman, Alexander F. Vakakis, Kathryn H. Matlack
+Department of Mechanical Science and Engineering, University of Illinois at Urbana Champaign
+1.
+ADDITIONAL INFORMATION FOR WAVENUMBER SCATTERING
+Fig. S1 provides a graphical illustration of the signal processing processes described in section II and Appendix B.
+Starting in the spatio-temporal domain, snap-shots of the wave velocity are taken successively and converted into the
+wavelet domain. This domain is partitioned into 12 bands (Fig. S1(b)). The inverse transformation of the k-th band
+partition gives at a fixed point in time gives the velocity vector ˙uk(x). The instantaneous energy of the k-th band
+is then conveniently computed as KE = 1
+2 ˙uT M ˙u or equivalently, KE = 1
+2
+�
+n ˙u2
+nmn. The energies are contacted
+over time to deliver the energy corresponding to wave propagation on the k-th band; note that minimal is shown for
+wave energy reconstruction when the sum of energy over all 12 partitions is compared to exact corresponding energy
+computed by direct numerical integration of the governing equations of motion.
+FIG. S1. Graphical illustration of the wavelet-based wavenumber partitioning processes.
+
+K12
+3
+K11
+K
+2.5
+K10
+Kg
+Wavenumber
+2
+K:
+K7
+1.5
+K6
+K5
+K4
+K10
+K4
+K:
+K5
+K11
+0.5
+K2
+Ki
+0
+K6
+K12
+100
+200
+300
+400
+500
+600
+Unit Cell No.Lotal3Energies Summary (Transformed Coords
+2.5
+Reconstruced Kinetic Energy
+-Physical Kinetic Energy
+Energy of Individual Partitions
+2 
+60
+nerg
+1
+0.5
+0
+0
+20
+40
+60
+80
+100
+20
+Error
+0
+%
+-20
+20
+40
+60
+80
+100
+Arbitrary TimeTime
+(ai)
+(aii)
+(aii)
+10-6
+(aiv)
+(av)
+(avi)
+Ki
+K2
+K3
+K4
+K5
+K6
+10-7
+10-8
+Time
+10-9
+(avii)
+aiix
+aix)
+(ax)
+(axi)
+(axii)
+K7
+K
+K10
+Kg
+K11
+10-10
+Position
+Position
+Position
+Position
+Position
+PositionK12
+3
+K11
+K
+2.5
+K10
+Kg
+Wavenumber
+2
+K:
+K7
+1.5
+K6
+K5
+K4
+K10
+K4
+K:
+K5
+K11
+0.5
+K2
+Ki
+0
+K6
+K12
+100
+200
+300
+400
+500
+600
+Unit Cell No.K12
+3
+K11
+K
+2.5
+K10
+Kg
+Wavenumber
+2
+K:
+K7
+1.5
+K6
+K5
+K4
+K10
+K4
+K:
+K5
+K11
+0.5
+K2
+Ki
+0
+K6
+K12
+100
+200
+300
+400
+500
+600
+Unit Cell No.K12
+3
+K11
+K
+2.5
+K10
+Kg
+Wavenumber
+2
+K:
+K7
+1.5
+K6
+K5
+K4
+K10
+K4
+K:
+K5
+K11
+0.5
+K2
+Ki
+0
+K6
+K12
+100
+200
+300
+400
+500
+600
+Unit Cell No.K12
+3
+K11
+K
+2.5
+K10
+Kg
+Wavenumber
+2
+K:
+K7
+1.5
+K6
+K5
+K4
+K10
+K4
+K:
+K5
+K11
+0.5
+K2
+Ki
+0
+K6
+K12
+100
+200
+300
+400
+500
+600
+Unit Cell No.K12
+3
+K11
+K
+2.5
+K10
+Kg
+Wavenumber
+2
+K:
+K7
+1.5
+K6
+K5
+K4
+K10
+K4
+K:
+K5
+K11
+0.5
+K2
+Ki
+0
+K6
+K12
+100
+200
+300
+400
+500
+600
+Unit Cell No.K12
+3
+K11
+K
+2.5
+K10
+Kg
+Wavenumber
+2
+K:
+K7
+1.5
+K6
+K5
+K4
+K10
+K4
+K:
+K5
+K11
+0.5
+K2
+Ki
+0
+K6
+K12
+100
+200
+300
+400
+500
+600
+Unit Cell No.2
+2.
+EXTENDED RESULTS FOR WAVENUMBER ENTROPY
+The results of Fig. 5 were recovered for the entire ensemble of simulations conducted for the diatomic (2-band)
+lattice of section II. The entire ensemble considered VI configurations depicted in Fig. 1 for excitation wavenumbers
+ranging from 2π/9 to 7π/9. The resulting normalized wavenumber entropy trends with respect to input forcing are
+given in Fig. S2 for all simulations, where it is seen that the trends presented in section II are agnostic to the excitation
+wavenumber. Power law fits are superimposed onto each subplot, and the adjusted R-squared values of the fits range
+between 0.9 and 0.99 for nearly every simulation.
+FIG. S2. Wavenumber entropy versus excitation amplitude for all datasets generated for the diatomic lattice system of section II.
+
+3
+3.
+DISPERSION BAND SELECTION FOR THE 4-BAND LATTICE
+Details on the dispersion band selection for the 4-band lattice considered in section III are provided in Fig. S3. The
+deflections of the Bloch-eigenmodes of the lattice were computed by solving the Bloch-eigenproblem over a sweep of
+waveumbers in the Irreducible Brillouin Zone (IBZ). Within a unit cell, the deflection of the resonator is computed
+as, w = |˜u2 − ˜u3|, of the Bloch-eigenmode in terms if of λ and η as stated in the main text: m1 = m4 = m = 0.005 kg
+and k1 = k4 = k = 2 × 104 N/m by m2 = m(1 − η), m3 = mη, and k3 = kλ while we fix k2 = 104 N/m. Note that the
+notation ˜u indicates displacement defined over the Bloch-eigenmode, not to be confused with the notation u which
+FIG. S3. Top: The deflections of the Bloch-eigenmodes for oscillators ˜u1-˜u4 of the 4-band lattice for each band, as well as
+w = |˜u3 − ˜u2| depicting total deflection of the resonator; bottom: The cost-function with respect to maximum deflection of the
+resonator on the second band (w) subject to out-of-phase motions, maximum group velocity, and a weighed measure considering
+both the deflection w and the group velocity; the red squares the optimal pairing of the parameters (λ,η), and the insets depict
+the resulting dispersion relations.
+
+4
+corresponds to coordinate displacements of the finite lattice in the main text. The Bloch-eigenmodes thus satisfy the
+following eigenvalue problem:
+�
+�
+�mω2
+�
+��
+1
+0
+0 0
+0 1 − η 0 0
+0
+0
+η 0
+0
+0
+0 1
+�
+�� − k
+�
+��
+3/2
+0
+0
+−1e−iκ
+−1/2 1 + λ −λ
+−1/2
+0
+−λ
+λ
+0
+−eiκ −1/2
+0
+3/2
+�
+��
+�
+�
+�
+�
+�
+�
+˜u1
+˜u2
+˜u3
+˜u4
+�
+�
+� = 0.
+(S1)
+This gives four Bloch-eigenmode solutions for x(κ) corresponding to the four bands of the lattice. The resonator of
+the 4 DoF model is described by ˜u2(κ) and ˜u3(κ). As explained in the main text, it is best that the second band
+corresponds to out-of-phase motion between these two coordinates, and that the deflection is maximized with respect
+to the system parameters. To maximize deflection subject to only out-of-plane motion, the signs of ˜u2 and ˜u3 are to be
+different, and hence this is recovered by maximizing |w|sign(−˜u2˜u3). The group velocities over the bands is considered
+as well by finding the maximum in the IBZ,yielding the use of the weighted measure, [max u]λ,η(vg|w|sign(−˜u2˜u3))
+where λ and η relate stiffnesses and masses in the unit cell.
+The cost-function recovered for deflection, group velocity, and the weighted measure between the two are graphically
+shown in S3, together with the dispersion that is recovered by selecting the optimal point in a parameter grid. The
+parameter pairing best suited for maximizing the previous weighted measure was taken as the ideal parameter settings
+to achieve inter-band energy transfers from low-to-high bands (section III). The grid approach was selected because
+the eigensolutions of Eq (S1) are too cumbersome to write-out analytically, and were not amenable for Newton-
+based straightforwardly. While a numerical scheme based on finite differences could resolve this, the search space
+was sufficiently confined and the problem was sufficiently small that direct grid search was not costly to perform.
+Moreover, the cost functions of Fig. S3 show trivial minimum and maximum solutions.
+
+5
+4.
+ADDITIONAL RESULTS FOR IBTET
+A.
+Recovered phase trajectories in the full lattice system
+The phase trajectories on branches of NNMs in the FEPs of the ROM reported in the main text (Fig. 11) revealed
+that the VI oscillator undergoes various dynamic regimes with varying energy, ranging from a low-energy linear system
+to a high energy smooth system governed by the elastic vibro-impact potential. The phase trajectories across regions
+I-IV of Fig. 11 can be compared to the corresponding phase plots of the full lattice in order to confirm that this
+physical mechanism is indeed seen in the lattice. To do this, simulations were considered whereby only one VI unit
+cell is embedded in the lattice with either Hertzian or bilinear contact law. The time series of the oscillators comprising
+the VI unit cell of the lattice were then considered, and phase trajectories could be recovered in the u1- ˙u1 and u2- ˙u2
+planes, where u1 corresponds to the outer mass of the unit cell and u2 to the inner mass (the VI resonator).
+Figs. S4 and S5 show the resulting phase portraits recovered for simulations of the full phononic lattice excited
+at various energies for both Hertzian and bilinear contact models, respectively. Low energy orbits are smooth and
+circular, indicating a linear response. Responses in the low-energy VI region (phase trajectory 2) are nearly the same,
+but with clear modulation and irregularity shown towards to origin of the host mass orbit (red), directly corresponding
+to the grazing region II of the FEP of the unit cell ROM. Higher-energy excitations (plots 3-4) in the fully VI energy
+regimes reveal non-smooth temporal dynamics, as predicted by region III of the unit cell FEP. Finally, high energy
+simulations result in phase trajectories that are nearly regular again, with motions of the host mass and resonator
+being in-phase and nearly completely overlaying each other indicating that the clearance now has nearly no effect,
+directly in correspondence of region IV of the unit cell FEP of the ROM.
+FIG. S4. The phase trajectories of the masses of a single VI unit cell obeying the Hertzian contact law embedded in a full
+lattice masses of a single VI unit cell obeying the Hertzian contact law, plotted for various energies (right panels), and the
+corresponding normalized IBTET with respect to input energy (left panel - red dots).
+
+6
+FIG. S5. The phase trajectories of the masses of a single VI unit cell obeying the bilinear contact law embedded in a full
+lattice masses of a single VI unit cell obeying the bilinear contact law, plotted for various energies (right panels), and the
+corresponding normalized IBTET with respect to input energy (left panel - red dots).
+
+7
+B.
+Detailed simulation response for bilinear system
+Fig. 7 of the main text depicts a graphical summary of computational and post-processing results for the 4-band
+lattice with embedded Hertzian VI nonlinearities. For completeness, Fig. S6 depicts the same computational summary
+computed for a system with embedded bilinear VI nonlinearity. The same remarks stated for Fig. 7 in the main text
+apply to Fig. S6 as well, further corroborating the similarities in behavior between Hertzian VIs and bilinear VIs with
+respect to IBTET.
+FIG. S6. IBTET in the 4-band lattice with bilinear VI nonlinearity and 5 VI sites: (a) shows the evolution of the propagating
+wave energy; (b-e) propagation of the wave energy corresponding to each band of the lattice based on the numerically recovered
+dispersion of the full simulation; (f,g) dispersion of the input and output segments (labeled in (a)) demonstrating the targeted
+energy transfer to the higher bands; (h,i) Fourier spectra corresponding to the velocity of the four unit cell DoFs selected before
+(5-th unit cell) and after (150-th unit cell) VI engagement, with the four band-pass regions depicted with shading and insets
+depicting the corresponding velocity time histories.
+
+8
+C.
+Influence of input bandwidth and number of VI
+To understand the effect that the forcing profile has on the results presented in section III, an additional set of
+simulations was performed subject to 15 cycles of input forcing instead of 30. The results are given in Fig. S7 where
+very similar trends to Fig. 12 are recovered.
+This indicates that the mechanisms for energy transfer are indeed
+non-resonant, as the duration of the oscillations that the VIs are subject to does not modify overall performance.
+Moreover, the effect of having only a single VI unit cell configuration is was considered as well. To this end, another
+set of simulations was performed subject to the 30 cycle excitation as the case for Fig. 12 of the main text, but now
+for only 1 VI embedded within the finite lattice. The resulting IBTET are given in Fig. S8 with the normalized FEP
+slope superimposed. The same trends are recovered again, but with some minor differences. The total energy transfer
+achievable is unsurprisingly less (maxing out at approximate 10 percent). Hence, the normalization constants for
+the FEP slopes are slightly different, which is why the FEP slopes superimposed appear slightly different in Fig. S8.
+Moreover, there are more pronounced perturbations from the smooth decay trends as compared to the 5 VI case,
+and this is due to the volatility of the non-resonant VI dynamics which are smoothed-out by incorporating more VIs.
+In other words, the energy transfer is dependent on the momentum transfer of incident waves. With additional VIs,
+this momentum transfer is better averaged out across the system as compared to the single VI case. However, the
+agreement in the overall trends of Fig. S8 supports the arguments developed in section IV for the evolution of the
+BTET mechanism with respect to system energy.
+FIG. S7. The same as Fig. 12, but for 15 cycles of input excitation instead of 30. The relative energy inter-band energy transfer,
+with the normalized slope from the ROM-FEP superimposed for (a) Hertzian and (b) bilinear contact models; the dashed lines
+depict the normalized FEP slopes, the gray lines depict the normalized FEP slopes lower-bounded by the initial (linear) energy
+of the higher bands, and green lines depict a power law fit to red dots, with the adjusted R-squared value shown with the inset.
+
+9
+FIG. S8. The same as Fig. 12, but for 1 VI engaged instead of 5. The relative energy inter-band energy transfer, with the
+normalized slope from the ROM-FEP superimposed for (a) Hertzian and (b) bilinear contact models; the dashed lines depict
+the normalized FEP slopes, the gray lines depict the normalized FEP slopes lower-bounded by the initial (linear) energy of the
+higher bands, and green lines depict a power law fit to red dots, with the adjusted R-squared value shown with the inset.
+

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@@ -0,0 +1,2032 @@
+Lepton Flavor Specific Extended Higgs Model
+B. L. Gon¸calves
+Departamento de F´ısica and CFTP, Instituto Superior T´ecnico,
+Universidade de Lisboa, Lisboa, Portugal and
+Centro de F´ısica Te´orica e Computacional,
+Faculdade de Ciˆencias, Universidade de Lisboa,
+Campo Grande, Edif´ıcio C8, 1749-016 Lisboa, Portugal
+Matthew Knauss and Marc Sher
+High Energy Theory Group, William & Mary,
+Williamsburg, VA 23187, USA
+(Dated: January 23, 2023)
+Abstract
+In extended Higgs models, a discrete symmetry is needed in the quark sector to avoid tree-level
+flavor-changing neutral currents. However, this is not necessary the case in the lepton sector. We
+consider a model in which one Higgs couples to quarks and three others couple to the electron,
+muon and tau, respectively. This four-doublet model is presented with the full scalar potential and
+the gauge and Yukawa couplings. The constraints from boundedness, perturbativity and oblique
+parameters are incorporated as well as constraints from meson-antimeson mixing, radiative B-
+decays and the diphoton Higgs decay rate. We also consider bounds from searches for heavy neutral
+and charged scalars at the LHC. Since the Standard Model Higgs couplings match predictions very
+well, we focus on the alignment limit of the model. It is shown that for a wide range of parameters,
+the lightest additional scalar, pseudoscalar and charged scalar can have substantial decays into
+electrons and muons (in contrast to the usual leptonic decays into taus). An interesting signature
+in the neutral sector would be the production, through vector boson fusion, of a pair of scalars,
+each of which decays into an electron or muon pair.
+1
+arXiv:2301.08641v1  [hep-ph]  20 Jan 2023
+
+I.
+INTRODUCTION
+The Higgs boson was initially discovered [1, 2] through its decay into gauge
+bosons. Since then, the coupling of the Higgs to third generation fermions
+has also been determined with increasing accuracy [3–7].
+However, while there is evidence [8] of the Higgs decay into muons, there
+remain large uncertainties and the discovery has not yet been made. This
+leads one to ask if there are viable models in which the muon and tau couple
+to different Higgs bosons. It is often claimed that models in which fermions
+of a given charge couple to different Higgs bosons contain tree-level flavor
+changing neutral currents (FCNC). However, the seminal papers of Glashow
+and Weinberg [9] and of Paschos [10] explicitly referred to the quark sector.
+As we will see, FCNC can be avoided in the lepton sector even if different
+leptons couple to different Higgs bosons.
+The first such model, called the muon-specific Two Higgs Doublet (2HDM)
+model, was developed by Abe, Sato and Yagyu [11] (ASY). They use a Z4
+symmetry, under which the muon and tau have different quantum numbers,
+and break this softly. The model has no tree-level FCNC and the Yukawa
+couplings for the muon and tau are no longer simply proportional to their
+masses with the proportionality coefficient being the same for all flavours:
+rather, the ASY model can substantially enhance or suppress the muon in-
+teractions of scalars relative to those with tau leptons. The purpose of their
+model was to attempt an explanation of the muon g-2 anomaly, and for the
+parameters they considered the dimuon coupling of the 125 GeV Higgs is not
+suppressed. Their model can address the g-2 anomaly, but only for a very
+2
+
+narrow region of parameter-space. A more detailed analysis was carried out
+in Ref. [12] where the phenomenology of the model was studied.
+The ASY muon specific 2HDM used a Z4 discrete symmetry in which the
+left-handed muon doublet and right-handed singlet have charge i and Φ1 has
+charge -1. All other fields have charge +1. This then has Φ1 coupling to
+muons and Φ2 coupling to all other fermions. Ivanov and Nishi have pointed
+out [13] that the actual symmetry group of the model is a softly broken Z2
+in which Φ1 and µR are negative and with a U(1) corresponding to muon
+number. This does not affect the ASY Lagrangian. In this model, the mass
+matrix of the charged leptons breaks into a 2 × 2 submatrix, corresponding
+to e − τ and a 1 × 1 corresponding to the muon. One might be concerned
+about how the PMNS matrix is generated if the muon and muon neutrino
+mass matrices decouple. However, even if the charged lepton and neutrino
+mass matrices are diagonal, one will still obtain a PMNS matrix using the
+see-saw (type 1) mechanism. The light neutrino mass matrix is then mij =
+(MD)ik(MN)−1
+kl (MD)lj where MD is the diagonal Dirac neutrino mass matrix
+and MN is the superheavy Majorana right-handed neutrino mass matrix.
+The latter is arbitrary and so the light neutrino mass matrix is not diagonal,
+leading to a non-trivial PMNS matrix. Note that this will not work in the
+quark sector.
+In this paper, we take the ASY model one step further and suppose that
+each of the charged leptons couples to a different Higgs doublet, which we
+will label as Φe, Φµ and Φτ. This can be achieved with a (Z4)e×(Z4)µ×(Z4)τ
+symmetry in which Lℓ and ℓR have quantum number under (Z4)ℓ of i and
+the Φℓ has quantum number −1. Equivalently, one can replace the Z4 with
+3
+
+Z2 × U(1) as discussed above - the Lagrangian in either case is identical. To
+achieve a non-trivial PMNS matrix, the symmetry must be softly broken in
+the superheavy Majorana neutrino mass matrix. The simplest implementa-
+tion of this model would be a 4HDM in which the fourth Higgs Φq couples
+to the quarks. This is similar to the lepton-specific model. Certainly one
+could have one of Φℓ be the same as Φq, leading to a 3HDM. However, if the
+Φq is Φτ, then the resulting model is very similar to the muon-specific model
+- the only difference being the very small interaction of the Higgs with the
+electron. For simplicity, we assume they are separate. One could also adopt
+a type-II structure, with 5HDM, but that brings in additional complications
+and the type-II parameter space is much narrower than the type-I. So, we
+will focus on the 4HDM with Φq, Φe, Φµ and Φτ.
+Although there are hundreds of papers that study models with three Higgs
+doublets, very few look at models with four. A recent paper with 4HDM in
+which each Higgs couples to sets of fermions with similar masses has been
+proposed [14] and a special ansatz, “singular alignment”, is needed to sup-
+press FCNC. A supersymmetric model [15] had one doublet each coupling
+to up-quarks, down-quarks and leptons, with the fourth needed for anomaly
+cancellation. A similar non-supersymmetric model was proposed [16](with
+the fourth Higgs needed to relax some tight constraints). An early discus-
+sion that mentions 4HDMs [17] studied Abelian symmetries in multidoublet
+models. There are also many studies of symmetries and vacuum states of
+N doublet models. An extremely extensive 2017 review of Ivanov [18], with
+over 500 references, studied numerous extended scalar sectors (including two
+doublet models, N doublet models, singlet and triplet extensions). Most rel-
+4
+
+evant papers before that time are referred to in this review. A more recent
+paper [19] looked at the interesting issue of non-decoupling in multiscalar
+models. Related work [20] dealt with large discrete symmetry groups in N
+doublet models. Additionally, the “Private Higgs” model of Porto and Zee
+[21, 22] had one Higgs doublet for every fermion. In contrast to the model we
+propose, their model had numerous discrete symmetries and included several
+“darkon” scalars.
+In section II, the model is presented, including the full scalar potential and
+the gauge and Yukawa couplings. In section III, we discuss the constraints
+on the potential from boundedness and constraints from oblique parameters.
+In section IV, two benchmark models are presented. In the first model, the
+potential is divided into two 2×2 subsections and in the second, the full 4×4
+model is discussed in the experimentally indicated alignment limit. Section
+V contains our results and conclusions.
+II.
+THE MODEL
+A.
+Scalar sector
+The potential can be written as a sum of quadratic and quartic terms:
+V = V2 + V4. We allow for soft breaking of the discrete symmetry in the
+quadratic terms:
+V2 = m2
+qqֆ
+qΦq + m2
+eeֆ
+eΦe + m2
+µµΦ†
+µΦµ + m2
+ττΦ†
+τΦτ
++ [m2
+qe(Φ†
+qΦe) + m2
+qµ(Φ†
+qΦµ) + m2
+qτ(Φ†
+qΦτ)
++ m2
+eµ(Φ†
+eΦµ) + m2
+eτ(Φ†
+eΦτ) + m2
+µτ(Φ†
+µΦτ)] + h.c.
+(1)
+5
+
+and
+V4 = λq
+1(Φ†
+qΦq)2 + λe
+1(Φ†
+eΦe)2 + λµ
+1(Φ†
+µΦµ)2 + λτ
+1(Φ†
+τΦτ)2
++ λqe
+3 (Φ†
+qΦq)(Φ†
+eΦe) + λqµ
+3 (Φ†
+qΦq)(Φ†
+µΦµ) + λqτ
+3 (Φ†
+qΦq)(Φ†
+τΦτ)
++λeµ
+3 (Φ†
+eΦe)(Φ†
+µΦµ) + λeτ
+3 (Φ†
+eΦe)(Φ†
+τΦτ) + λµτ
+3 (Φ†
+µΦµ)(Φ†
+τΦτ)
++ λqe
+4 (Φ†
+qΦe)(Φ†
+eΦq) + λqµ
+4 (Φ†
+qΦµ)(Φ†
+µΦq) + λqτ
+4 (Φ†
+qΦτ)(Φ†
+τΦq)
++ λeµ
+4 (Φ†
+eΦµ)(Φ†
+µΦe) + λeτ
+4 (Φ†
+eΦτ)(Φ†
+τΦe) + λµτ
+4 (Φ†
+µΦτ)(Φ†
+τΦµ)
++ 1
+2
+�
+λqe
+5 (Φ†
+qΦe)2 + λqµ
+5 (Φ†
+qΦµ)2 + λqτ
+5 (Φ†
+qΦτ)2
++ λeµ
+5 (Φ†
+eΦµ)2 + λeτ
+5 (Φ†
+eΦτ)2 + λµτ
+5 (Φ†
+µΦτ)2 + h.c.
+�
+(2)
+Here, we have labeled the quartic couplings to be similar to the standard
+2HDM potential.
+We can write the Higgs bosons as
+Φi =
+�
+�
+φ+
+i
+(vi + φi + iχi)/
+√
+2
+�
+� , (i = q, e, µ, τ)
+(3)
+where the vi/
+√
+2 are the vacuum values of the neutral components. To discuss
+diagonalizing mass matrices and the various angles involved, we follow the
+procedure of Boto, Rom˜ao and Silva [23] closely.
+Without loss of generality, we can define the angles that rotate the fields
+into the Higgs basis in which only one scalar field gets a vev by
+vq = v cos β2 cos β3 cos β4
+ve = v sin β2 cos β3 cos β4
+vµ = v sin β3 cos β4
+vτ = v sin β4
+(4)
+6
+
+giving
+�
+�
+�
+�
+�
+�
+�
+�
+h0
+H1
+H2
+H3
+�
+�
+�
+�
+�
+�
+�
+�
+= Oβ
+�
+�
+�
+��
+�
+�
+�
+�
+φq
+φe
+φµ
+φτ
+�
+�
+�
+�
+�
+�
+�
+�
+(5)
+where
+Oβ =
+�
+�
+�
+�
+�
+�
+�
+�
+cβ2cβ3cβ4
+sβ2cβ3cβ4
+sβ3cβ4
+sβ4
+−sβ2
+cβ2
+0
+0
+−cβ2cβ3
+−sβ2sβ3
+cβ3
+0
+−cβ2cβ3sβ4 −sβ2cβ3sβ4 −sβ3sβ4 cβ4
+�
+�
+�
+�
+�
+�
+�
+�
+(6)
+Here, h0 is the field that gets the entire vev, v, and cθ (sθ) are cos θ (sin θ).
+From this basis, we can now diagonalize the mass matrices of the various
+scalars. In the neutral scalar sector, the physical neutral Higgs masses are
+given by
+�
+�
+�
+�
+�
+�
+�
+�
+h1
+h2
+h3
+h4
+�
+�
+�
+�
+�
+�
+�
+�
+= Oα
+�
+�
+�
+�
+�
+�
+�
+�
+φq
+φe
+φµ
+φτ
+�
+�
+�
+�
+�
+�
+�
+�
+(7)
+where h1 is the 125 GeV Higgs particle. For Oα, we use
+Oα = R34R24R23R14R13R12
+(8)
+Here, for example, R24 is given by
+R24 =
+�
+�
+�
+�
+�
+�
+�
+�
+1
+0
+0
+0
+0
+cα24
+0 sα24
+0
+0
+1
+0
+0 −sα24 0 cα24
+�
+�
+�
+�
+�
+�
+�
+�
+(9)
+7
+
+and the other R matrices follow. We see that there are six rotation angles.
+In the pseudoscalar sector, one has
+�
+�
+�
+�
+�
+�
+�
+�
+G0
+A1
+A2
+A3
+�
+�
+�
+�
+�
+�
+�
+�
+= OγOβ
+�
+�
+�
+�
+�
+�
+�
+�
+χq
+χe
+χµ
+χτ
+�
+�
+�
+�
+�
+�
+�
+�
+(10)
+where Oγ = P34P24P23 and, as before, for example
+P24 =
+�
+�
+�
+�
+�
+�
+�
+�
+1
+0
+0
+0
+0
+cγ24
+0 sγ24
+0
+0
+1
+0
+0 −sγ24 0 cγ24
+�
+�
+�
+�
+�
+�
+�
+�
+(11)
+Note that there are only three matrices here, since the Goldstone boson
+direction is fixed.
+Finally, in the charged sector
+�
+�
+�
+�
+�
+�
+�
+�
+G+
+H+
+1
+H+
+2
+H+
+3
+�
+�
+�
+�
+�
+�
+�
+�
+= OδOβ
+�
+�
+�
+�
+�
+�
+�
+�
+φ+
+q
+φ+
+e
+φ+
+µ
+φ+
+τ
+�
+�
+�
+�
+�
+�
+�
+�
+(12)
+where Oδ = Q34Q24Q23 and, as before, for example
+Q24 =
+�
+�
+�
+�
+�
+�
+�
+�
+1
+0
+0
+0
+0
+cδ24
+0 sδ24
+0
+0
+1
+0
+0 −sδ24 0 cδ24
+�
+�
+�
+�
+�
+�
+�
+�
+(13)
+8
+
+B.
+Gauge and Yukawa couplings
+1.
+Gauge couplings
+The scalar kinetic Lagrangian, Lk, defined as
+Lk =
+4
+�
+i=1
+|DµΦi|2
+(14)
+with the usual expression for the covariant derivative Dµ, contains the terms
+relevant to obtain the trilinear couplings of the scalars and gauge bosons.
+The couplings ZZhi and W ±W ∓hi are written in the form
+� 4
+�
+i=1
+Cihi
+� � g
+2cW
+mZZµZµ + gmWW −
+µ W +µ
+�
+.
+(15)
+The Ci factors are included in Appendix A. It is possible to check that, when
+the set of conditions α1j = βj is verified (for j = 2, 3, 4), one gets C1 = 1
+together with Ck = 0, for k ̸= 1, which defines the alignment limit in this
+model.
+2.
+Yukawa couplings
+Following the notation of Branco, et al. [24], the couplings of the scalar
+and pseudoscalar Higgs are defined through
+LS
+Y = −
+�
+f∈{q,e,µ,τ}
+mf
+v
+�
+ξf
+h1 ¯ffh1 + ξf
+h2 ¯ffh2 + ξf
+h3 ¯ffh3 + ξf
+h4 ¯ffh4
+�
+LP
+Y = −
+�
+f∈{q,e,µ,τ}
+�
+−imf
+v
+� �
+ξf
+A1 ¯fγ5fA1 + ξf
+A2 ¯fγ5fA2 + ξf
+A3 ¯fγ5fA3
+�
+(16)
+9
+
+where ξf
+hj and ξf
+Aj are given by
+ξq
+hj = Oαj,1
+ˆv1
+, ξe
+hj = Oαj,2
+ˆv2
+, ξµ
+hj = Oαj,3
+ˆv3
+, ξτ
+hj = Oαj,4
+ˆv4
+ξq
+Aj =
+(OγOβ)j,1
+ˆv1
+, ξe
+Aj =
+(OγOβ)j,2
+ˆv2
+, ξµ
+Aj =
+(OγOβ)j,3
+ˆv3
+, ξτ
+Aj =
+(OγOβ)j,4
+ˆv4
+(17)
+using ˆvi ≡ vi/v. Similarly, the couplings of the charged Higgs are defined
+through
+LC
+Y = −
+�
+j
+� �
+u,d
+√
+2Vud
+v
+¯u
+�
+muξqL
+H+
+j PL + mdξqR
+H+
+j PR
+�
+dH+
+j
++
+�
+l
+√
+2ml
+v
+ξlL
+H+
+j ¯νLlRH+
+j
+�
+(18)
+where ξf
+H+
+j are given by
+ξqLR
+H+
+j
+=
+(OδOβ)j,1
+ˆv1
+, ξeL
+H+
+j =
+(OδOβ)j,2
+ˆv2
+, ξµL
+H+
+j =
+(OδOβ)j,3
+ˆv3
+, ξτL
+H+
+j =
+(OδOβ)j,4
+ˆv4
+(19)
+A table of general Yukawa couplings are included in Appendix B.
+III.
+THEORETICAL CONSTRAINTS ON THE SCALAR POTENTIAL
+A.
+Bounded from below constraints
+In extensions of the scalar sector, one needs to choose quartic parame-
+ters such that the potential is bounded from below (BFB)1. While this is
+straightforward in the 2HDM, it can be quite complicated in models with
+1 We require that the potential be bounded at scales where the quartic terms dominate. The case in which
+the potential turns over at very high scales due to renormalization group running will not be considered.
+In fact, the Standard Model itself would not satisfy that latter condition
+10
+
+more than two doublets. An added complication in models with doublets is
+that there can be an instability in the charged scalar direction even if there
+is stability in the neutral scalar direction (see Ref. [25] for an example). A
+recent discussion of these conditions for a three-doublet model can be found
+in the work of Boto, Rom˜ao and Silva [26]. They showed that while necessary
+and sufficient conditions are known for the neutral direction, only sufficient
+conditions are known for stability in the charged direction, and they discuss
+a general strategy. We will first discuss the neutral directions.
+Looking at the neutral direction, the 2HDM potential can be written as
+V4 = a11H4
+1 + a22H4
+2 + a12H2
+1H2
+2, where the matrix is symmetric. The con-
+ditions for copositivity (where the potential is positive for all values of H2
+1
+and H2
+2) are given by a11 ≥ 0, a22 ≥ 0, a12 + √a11a22 ≥ 0. As shown in
+Refs. [27, 28], for the neutral sector of the 3HDM, the conditions are
+a11 ≥ 0,
+a22 ≥ 0,
+a33 ≥ 0
+(20)
+a12 + √a11a22 ≥ 0
+(21)
+a13 + √a11a33 ≥ 0
+(22)
+a23 + √a22a33 ≥ 0
+(23)
+√a11a22a33 + a12
+√a33 + a13
+√a22 + a23
+√a11 ≥ 0
+(24)
+det A ≥ 0
+(25)
+where A is the matrix with entries aij. Clearly, the first line is needed for
+stability along the axes, the next three lines are needed for stability in the
+three planes, and the last two lines ensure stability for all directions. For the
+4HDM that we consider, the corresponding conditions must be satisfied for
+every three dimensional subspace. The remaining conditions are extremely
+11
+
+complicated, but are given in full in Ref. [27]. We have incorporated the
+conditions in that paper to ensure stability in the neutral directions.
+As shown by Boto, Rom˜ao and Silva [26], even in the 3HDM there are no
+straightforward necessary and sufficient conditions for stability in the charged
+directions. In the 2HDM, with a quartic potential
+V4 = λ1(Φ†
+1Φ1)2+λ2(Φ†
+2Φ2)2+λ3(Φ†
+1Φ1)(Φ†
+2Φ2)+λ4|Φ†
+1Φ2|2+1
+2λ5[(Φ†
+1Φ2)2+(Φ†
+2Φ1)2]
+(26)
+the condition for stability is [29, 30] λ3 + λ4 − |λ5| ≥ −2√λ1λ2. Rather than
+attempt a detailed numerical study of stability in the 4HDM case, we will
+require that this condition be satisfied for all 2 × 2 subspaces of the 4HDM.
+This requirement is, of course, necessary but may not be sufficient.
+B.
+Oblique Parameters
+To discuss the S, T, U oblique parameters, we follow the methods and
+results in Grimus, et al [31]. To do this, we can write the matrices ˜U and
+˜V from Grimus, et al [31] using our notation in the previous section. ˜V is
+defined through
+�
+�
+�
+�
+�
+�
+�
+�
+φ1 + iχ1
+φ2 + iχ2
+φ3 + iχ3
+φ4 + iχ4
+�
+�
+�
+�
+�
+�
+�
+�
+= ˜V
+�
+h1 h2 h3 h4 G0 A1 A2 A3
+�T
+(27)
+where
+˜V ≡
+�
+�
+O−1
+α
+i (OγOβ)−1
+�
+�
+(28)
+12
+
+Notice in Eq. (27), our notation slightly differs from Grimus et al [31] by
+keeping the Goldstone boson with the pseudoscalar mass eigenstates.
+˜U is defined as
+�
+�
+�
+�
+�
+�
+�
+�
+φ+
+1
+φ+
+2
+φ+
+3
+φ+
+4
+�
+�
+�
+�
+�
+�
+�
+�
+= ˜U
+�
+�
+�
+�
+�
+�
+�
+�
+G+
+H+
+1
+H+
+2
+H+
+3
+�
+�
+�
+�
+�
+�
+�
+�
+(29)
+where
+˜U ≡
+�
+OδOβ
+�
+(30)
+We take the values of S, T from [32] with
+S = −0.02 ± 0.10
+T =
+0.03 ± 0.12
+(31)
+We will not include the detailed calculation of the unitarity and perturba-
+tivity bounds, due to the large number of scalar couplings. Rather, we will
+simply require that all of the quartic scalar couplings be less than 4π.
+IV.
+BENCHMARK MODELS
+As is clear from examining the scalar potential and the Appendices, the
+model contains a large number of free parameters. To focus on the most
+important aspects of the model, we will consider two benchmark models. In
+the first, we will assume that the (qτ) sector of the Higgs potential decouples
+from the (µe) sector. In that case, the 4 × 4 scalar mass matrices decouple
+into two 2 × 2 matrices which can be trivially diagonalized analytically. In
+the second benchmark model, we will take the alignment limit. In the con-
+ventional 2HDMs, this is equivalent to cos(α − β) = 0, with tan β ≡ v2/v1
+13
+
+and α diagonalizes the scalar mass matrix. This limit is often chosen since it
+means that the couplings of the 125 GeV Higgs boson are identical to that
+in the Standard Model (which seems to be preferred by LHC data). In this
+case, it is easy to see from Appendices A and B that the alignment limit
+corresponds to α1j = βj, as previously stated. Since the coupling of the 125
+GeV Higgs is the same as the Standard Model, there is no need to study
+Higgs production and tree-level decays in this case.
+A.
+The Model without (qτ)-(µe) mixing
+In this model, the absence of (qτ)-(µe) mixing means that the matrix that
+diagonalizes the scalar mass matrix, Oα, is broken into two 2 × 2 matrices.
+The upper 2 × 2 matrix looks very similar to the lepton-specific 2HDM. The
+only difference involves the coupling to the muon, which is not well-measured.
+However in this case, unlike the lepton-specific model, the value of v2
+q + v2
+τ is
+not v2 = (246 GeV)2 but will be smaller. As a result, all Yukawa couplings
+will be increased. This will affect the decays of the 125 GeV Higgs boson as
+well as the production.
+We define the parameter µX as
+µX ≡
+σ(pp → H)BR(H → X)
+σ(pp → H)SMBR(H → X)SM
+(32)
+and look at X = gg, µµ, ττ, ¯cc,¯bb, ¯tt, γγ, γZ, WW, ZZ. The results are in
+Figure 1, where we have plotted, in the usual way for 2HDMs, the allowed
+region in the tan β − cos(β − α) plane. We require all µX to be consistent
+with unity within 20% at 95% CL, which is a rough approximation to the
+14
+
+100%
+95%
+90%
+85%
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.05
+0.10
+0.50
+1
+5
+10
+50
+cos(β-α)
+tan(β)
+FIG. 1: Allowed regions in the tan β − cos (β − α) plane, in the model without (qτ)-(µe)
+mixing, for different values of r ≡
+�
+v2
+q + v2
+τ
+�1/2 /v, namely r = 1 in orange, r = 0.95 in
+purple, r = 0.90 in blue and r = 0.85 in cyan.
+precision of current data. 2
+We see that if the ratio of (v2
+q +v2
+τ)1/2 to v is less than 0.85, that the entire
+parameter space practically disappears. Thus much of the vev is saturated
+by vq and vτ. Clearly the coupling here to the muon vanishes and thus in
+the full model, the muonic decay of the Standard Model Higgs, if confirmed,
+will be a strong constraint.
+The shrinking of the parameter-space in the cos(β − α) < 0 allowed re-
+gion occurs mainly due to the combination of g2
+HV V , measured from Higgs
+production, and g2
+Hll, measured from Higgs decay.
+The shrinking of the
+parameter-space in the cos(β − α) > 0 allowed region mainly occurs due to
+2 We are looking in the context of the lepton-specific 2HDM - but now the combination of vacuum values,
+(v2
+q + v2
+τ)1/2 no longer is equal to the Standard Model vacuum value, v.
+15
+
+g2
+HQQ, from Higgs production, now combined with both g2
+Hqq and g2
+Hll.
+In itself, this benchmark model is phenomenologically unacceptable. Each
+2 × 2 submatrix will have a zero eigenvalue in the pseudoscalar and in the
+charged scalar sectors, leading to two zero eigenvalues in each sector. Only
+one can be absorbed by the W and Z gauge bosons. The additional massless
+scalars arise due to an additional accidental SU(2) symmetry. Thus, there
+must be some off-diagonal terms. We can include these terms but assume
+they are small and do a perturbative expansion.
+For simplicity, let us add a single off-diagonal term, λqµ
+5 . This will allow for
+nonzero masses for the lightest charged and pseudoscalar Higgs3. This term
+will modify the Yukawa couplings of the Standard Model 125 GeV Higgs.
+For the couplings of the quarks, for example, the Yukawa coupling gY ¯qqΦq
+is
+√
+2mq/vq.
+Writing Φq = V11h1 + V12h2 + ..., where h1 is the 125 GeV
+Higgs, one sees that the coupling is modified by a factor of
+v
+vqV11. One can
+perturbatively calculate the eigenvalues and eigenvectors of the mass matrix
+and we find that
+V11 = 1 − 1
+2ϵ2
+1
+� �
+c34s12
+m2
+h1 − m2
+h3
+�2
++
+�
+c12s34
+m2
+h1 − m2
+h4
+�2 �
+,
+(33)
+where ϵ1 = λqµ
+5 vqvµ, cij = cos αij (sij = sin αij) and the masses are the masses
+of the neutral scalars. The relevant point here is that V11 is reduced, which
+counters the effect of the smaller vq. In order for the lightest charged Higgs
+to have an acceptable mass, there is a minimum value of λqµ
+5 , but the masses
+of the neutral scalars can be large enough that the reduction (proportional
+3 One can decouple the masses of the charged and pseudoscalar Higgs by adding a λqµ
+4
+term and can easily
+satisfy any BFB concerns with a λqµ
+3
+term.
+16
+
+to (vµ/mh3)2) is quite small.
+B.
+The Aligned Model
+The full 4HDM has a large number of parameters in the scalar potential:
+10 quadratic terms and 22 quartic terms. Not surprisingly, many of these
+parameters will have little effect on phenomenology. As noted earlier, the
+fact that the 125 GeV Higgs has decays consistent with the Standard Model
+implies that multi-doublet models must be near the alignment limit in which
+the Standard Model Higgs interactions are unaffected. From Appendix A,
+we see that this will occur if α1j = βj. Parameters that might be of phe-
+nomenological relevance are then the βj, α23,24,34, the three γ parameters, the
+three δ parameters, the four scalar masses, the three charged masses and the
+three pseudoscalar masses, in addition to the SM Higgs vev. Instead of the
+potential’s couplings, we can choose to describe the model in terms of the
+previously mentioned parameters and six additional parameters, namely the
+remaining six m2
+ij, giving a total of 29 parameters4. As we will see, many of
+these parameters will not be relevant for particular processes.
+Choosing values for the rotation angles and the squared masses, it is pos-
+sible to define the scalar, pseudoscalar, and charged squared-mass matrices
+as M 2
+s,p,c = R−1Ds,p,cR, considering the corresponding R matrix for each case
+and D as the diagonal matrix with the squared masses in its entries. The
+quartic parameters of the Lagrangian can be expressed in terms of elements
+4 With the addition of the three α parameters which are defined through the alignment limit, we get 32
+parameters, just like the scalar potential.
+17
+
+of such matrices, the vevs and the m2
+ij parameters as the following:
+λi
+1 = 1
+2v3
+i
+�
+�viM 2
+s,ii +
+�
+j̸=i
+vjm2
+ij
+�
+� ,
+λij
+3 = 1
+vivj
+�
+M 2
+s,ij − 2M 2
+c,ij + m2
+ij
+�
+,
+λij
+4 = 1
+vivj
+�
+2M 2
+c,ij − M 2
+p,ij − m2
+ij
+�
+,
+λij
+5 = 1
+vivj
+�
+M 2
+p,ij − m2
+ij
+�
+,
+(34)
+in which i, j = q, e, µ, τ. In the 2HDM limit, these equations give rise to
+the well-known expressions for the λ parameters in terms of masses, angles,
+the electroweak vev v and the soft-breaking terms m2
+ij [24, 33]. For every
+possible set of parameters, we require the following:
+• The bounded-from-below conditions are satisfied.
+• The perturbativity condition that the absolute values of λ parameters
+are less than 4π is maintained.
+• The previous condition also applies to Yukawa couplings.
+• The values of the S and T parameters are within the range given by
+Eq. (31).
+• Charged Higgs masses must exceed 80 GeV [34].
+• Contributions from the charged scalars to the loop-induced Higgs dipho-
+ton decay h → γγ are compatible with experimental bounds.
+This
+is achieved by checking the value of the diphoton signal strength
+µγγ [35, 36] for each set of parameters.
+18
+
+• Bounds coming from new physics contributions to B meson oscillations,
+∆MBd,s, as well as K mesons, ∆MK, are within the experimental allowed
+range for each case [32, 37]. Such nonstandard contributions come from
+charged scalars through one-loop processes [38, 39].
+• Contributions to b → sγ [39], again from charged Higgs particles, are
+acceptable. In the Type II 2HDM, this gives the strongest constraint
+on charged Higgs bosons.
+• At the LHC, CMS [40] has searched for a heavy neutral Higgs decaying
+into τ pairs. Although done in the context of the MSSM, the results are
+very similar in this model (with adjusted Yukawa couplings, of course)
+and the production cross-section times branching ratio varies from 10
+pb to 10 fb over the range of masses from 150 GeV to 1000 GeV. More
+recently, ATLAS [41] has done a similar analysis. Note that one usually
+assumes that the decay into top quarks will dominate for masses above
+350 GeV, but that might not be the case here due to the lepton-specific
+nature of the model.
+We impose these experimental bounds on our
+parameter-space, which, up to small differences due to form factors,
+apply to neutral scalars and pseudoscalars.
+• Finally, we can consider LHC direct searches for heavy charged Higgs
+bosons. Searches fall into two categories - those in which the charged
+Higgs mass is greater than mt + mb and those in which it is less.
+– If it is greater, then the predominant decay mode will be into t¯b, ex-
+cept for the narrow window of parameter-space in which the charged
+19
+
+Higgs in question has essentially zero overlap with Φq. The produc-
+tion cross-section for a charged Higgs mass of 200, 300, 600 GeV
+is [42] within a factor of 2 (scaling the Yukawa coupling appropri-
+ately to a lepton-specific or Type I model) of 0.4, 0.1, 0.01 picobarns.
+ATLAS [43] has found bounds from Run II on the product of the
+production rate and the H+ → t¯b branching ratio. Their result is
+below our production cross-section by a factor of a few, and thus the
+model is not yet constrained by the non-observation at the LHC.
+– If the charged Higgs is lighter, then a major decay mode is into τντ.
+In this case the predominant production mode is through t → bH+.
+Since top production is well understood, searches at ATLAS [44]
+and CMS [45] place bounds on BR(t → bH+)BR(H+ → τντ). This
+bound may not be too restrictive, since a charged Higgs that is
+either quarkphobic or leptophobic will not contribute and thus it
+will depend on mixing angles. Nonetheless, we have incorporated
+the results of these searches in bounding our parameter-space.
+We will primarily focus on the lightest neutral scalar (other than the 125
+GeV Higgs), the lightest pseudoscalar and the lightest charged scalar. Re-
+sults from these scalars will also apply to the heavier scalars by appropriate
+choice of mixing angles (with the exception of heavy scalar decays into lighter
+scalars, which we will not consider). The lepton-specific 2HDM has one scalar
+coupling to quarks and another to leptons. The primary difference between
+our model and the lepton-specific model is that different scalars couple to the
+muon and the electron (note that the muon-specific model [11, 12] has the
+20
+
+same scalar coupling to the quarks and the τ, which is more like an extension
+of the type I 2HDM). As a result, we will focus on decays involving muons
+and electrons.
+We first consider the decay of the lightest neutral scalar (other than the
+125 GeV Higgs, which has Standard Model couplings in the alignment limit)
+into electrons, muons and taus.
+Since the heavier masses aren’t relevant
+in the analysis, the parameter-space is substantially reduced. We consider
+two mass regions, in which the scalar mass is below and above 350 GeV,
+respectively. In the latter case, decays to top quarks can be substantial, even
+if the mixing angles are small.
+As noted above, given the masses, soft-breaking mass parameters and mix-
+ing angles, the quartic couplings are determined. We scan the full parameter
+space and check each of the conditions above. Typically, we find several mil-
+lion parameter sets that are acceptable. The results are plotted in Figure 2.
+Note that in the Standard Model the branching ratio of the dimuon decay of
+the Higgs is 2 × 10−4 and this level (and somewhat below) is certainly exper-
+imentally accessible. One can see that for a scalar mass below 350 GeV, the
+dielectron decay branching ratio can be much, much larger than the Standard
+Model and the dimuon decay branching ratio can approach unity. Above 350
+GeV, the opening of the top decay channel, even if the mixing angle is very
+small, substantially reduces the leptonic branching ratios.
+It is not surprising that this can occur.
+If one chose parameters such
+that there was no mixing at all between Φee and the other scalars, then the
+only decay of the Φee would be into electrons. This would require extreme
+fine-tuning, since no symmetry will eliminate mixing in the quartic sector of
+21
+
+FIG. 2: These scatterplots show allowed points for h2 decays. Results are shown for h2
+masses below 350 GeV and above that mass scale (at which point the ¯tt channel opens
+up). The upper figures plot ee and µµ decays and the lower figures plot µµ and ττ decays.
+The decay branching ratio of the SM Higgs to µµ is approximately 2 × 10−4.
+the potential and even very small values of the quartic mixing terms would
+allow for other decays that could dominate. Nonetheless, we see many sets
+of parameters for which the dielectron and dimuon decays of this lightest
+neutral scalar (other than the Standard Model Higgs) can be substantial.
+In Figure 2, we also show the branching ratios to muons and to taus.
+Again, one can see that the absolute branching ratio to dimuons can be
+substantially more than that into two taus. Thus, we find that searches for
+heavy neutral Higgs bosons decaying into leptons, which generally focus on
+22
+
+10-1
+10-1
+mh² < 350 GeV
+mh, > 350 GeV
+10-2.
+10-2
+ee)
+BR(h2 →ee)
+10-3.
+10-3.
+BR(h2 →(
+10-4
+10-4.
+10-5.
+10-5.
+10-6.
+10-6
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+BR(h2 →μμ)
+BR(h2 →μμ)100
+100
+mh² < 350 GeV
+mh² > 350 GeV
+10-1
+10-1.
+10-2
+10-2.
+(nn
+(nn←
+个
+BR(h2)
+10-3
+10-4.
+10-4
+10-5,
+10-5-
+10-6.
+10-6.
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+BR(h2 → TT)
+BR(h2 → TT)FIG. 3: These scatterplots show allowed points for H± decays. Results are shown for H±
+masses below 180 GeV and above that mass scale (at which point the ¯tb channel opens
+up). The upper figures plot eν and µν decays and the lower figures plot µν and τν decays.
+tauonic decays, should also study muonic and electronic decays.
+Since we are in the alignment limit, there is no three-point coupling of
+these scalars to two gauge bosons. They could be produced in a collider
+through WW or ZZ fusion to two Φs. The signature would be two electron-
+positron or muon pairs each coming from a Φ. The electron-positron pair
+rate will be smaller, but more distinctive. While four lepton events have
+been searched for [46], we know of no analysis of this particular signature.
+An approximate production cross-section can be obtained by comparison
+with the inert doublet model[47] which has a similar production process.
+23
+
+100
+100
+mH < 180 GeV
+mH# > 180 GeV
+10-1.
+10-1.
+10-2.
+e=
+et,
+10-3.
+个
+10-3
+R
+10-4.
+10-
+B
+10-5.
+10-5.
+10-6.
+10-6.
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+BR(H→μvμ)
+BR(H→μvμ)100
+100
+10-1.
+10-1.
+>10-2.
+ 10-2.
+个
+10-3.
+10-3.
++1
+BR
+5 10-4.
+10-4.
+10-5.
+10-5.
+mH# < 180 GeV
+mH± > 180 GeV
+10-6.
+10-6.
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+BR(H→)
+BR(H→v)Typical production cross-sections at the LHC are approximately 0.5 fb. With
+an integrated luminosity of 3 ab−1, this means that branching fractions of
+O(10−3) or less will be difficult to detect until the next generation colliders.
+We have also studied the decays of the pseudoscalar into leptons and find
+very similar results. For the charged Higgs decays, we show the ratio of eν
+to µν decays as well as the individual branching ratios in Figure 3 as well as
+the µν to τν decays . Here, we consider mass ranges below and above 180
+GeV, at which point the t¯b opens up. Note that there are more points in
+the region above 180 GeV since below that mass a much higher proportion
+of points are experimentally excluded. There is a large number of points in
+which the electronic decays are substantial and the muonic decay branching
+ratios can approach unity.
+In Appendix C we show several benchmark points. These points satisfy
+all of the various constraints listed earlier in this section. For point S1, one
+can see that the h2 → µµ branching ratio is almost 47% and the electronic
+branching ratio is over 0.25%. Clearly, the signature would most likely be
+two muon pairs, each coming from a neutral scalar, most of the other decays
+being tau pairs or ¯bb, with an occasional electron-positron pair. In benchmark
+point S2, the dimuon decay of the scalar is smaller than that of the electron.
+Here, one would see the ditau decays dominate, but the electron-positron
+decays might be measurable.
+We also see some benchmark points for the lightest charged Higgs, looking
+at the region in which the mass is below 180 GeV so the top-bottom channel
+is not available. For point C1, the decay into muons is slightly bigger than
+the decay into taus, and the electronic decay is 0.2%. For C2, the muon
+24
+
+decay is the smallest and the electron decay is as high as 1.7%. Again, this
+shows that decays into muons and electrons might be much, much higher
+than in traditional 2HDMs.
+V.
+CONCLUSION
+It is often believed that all fermions of a given charge must couple to
+the same Higgs multiplet in order to avoid tree-level flavor-changing neutral
+currents. However this is only true in the quark sector and need not be true
+in the lepton sector. The quark mass matrix cannot be diagonal without
+eliminating CKM mixing, however the lepton mass matrix can be diagonal,
+since PMNS mixing can cover from the superheavy Majorana neutrino sector.
+We have studied a 4HDM in which one scalar doublet couples to quarks and
+the other three couple to the electron, muon and tau families, respectively.
+There are numerous constraints on such a model, including bounded from
+below constraints, perturbativity, S and T parameters, the diphoton decay
+of the Higgs, limits from meson-antimeson oscillations, radiative b decays
+and various LHC constraints from heavy scalar searches. Scanning the pa-
+rameter space, we find numerous acceptable points in which the dielectron
+and dimuon decays of the lightest neutral scalar (other than the 125 GeV
+Higgs) can be much, much larger than expected. The results for the lightest
+pseudoscalar and charged scalar are also presented.
+Generally, searches for heavier Higgs bosons focus (in the lepton sector)
+on decays into τs. However, this model shows that decays into electrons and
+muons can be substantial (and certainly easier to detect). An interesting
+signature at either a linear collider or a hadron collider arises from vector
+25
+
+boson fusion into two such Higgs bosons, each of which decays into an electron
+or muon pair. We know of no bounds on such a process and hope to see
+searches in the near future.
+Acknowledgments
+The work of MS and MK was supported by the National Science Foun-
+dation under Grant PHY-1819575.
+The work of BLG is supported by
+Funda¸c˜ao para a Ciˆencia e a Tecnologia (FCT, Portugal) through the
+PhD grant SFRH/BD/139165/2018 and the projects UIDB/00777/2020,
+UIDP/00777/2020,
+UIDB/00618/2020,
+UIDP/00618/2020,
+CERN/FIS-
+PAR/0019/2021 and CERN/FIS-PAR/0025/2021.
+BLG thanks the Ful-
+bright Commission in Portugal and William & Mary for support. MK thanks
+Pitt-PACC at the University of Pittsburgh for their hospitality. We thank
+Igor Ivanov for clarifying the symmetry group of the model, Arnab Dasgupta
+for coding help and suggestions and for useful discussions, and Pedro Ferreira
+for a helpful discussion of the lepton-specific 2HDM.
+26
+
+Appendix A: Gauge Couplings
+Trilinear Gauge Couplings ZZhi and W ±W ∓hi
+C1
+c12c13c14c2c3c4 + c13c14c3c4s12s2 + c14c4s13s3 + s14s4
+C2
+−c12c2c3c4(c24s13s23 + c13s14s24) − c23c24c3c4s12−2 − c24c3c4s12s13s23s2 −
+c13c3c4s12s14s24s2 + c13c24c4s23s3 − c4s13s14s24s3 + c14s24s4
+C3
+−c12c3c4[c13c24c2s14s34 + s23(−c2s13s24s34 + c34s2) + c23(c34c2s13 +
+s24s34s2)] + c34c4[c2c3s12s23 + c23(−c3s12s13s2 + c13s3)] +
+s34[c23c2c3c4s12s24 + c3c4s12s13s23s24s2 − c24c4s13s14s3 −
+c13c4(c24c3s12s14s2 + s23s24s3) + c14c24s4]
+C4
+−c2c3c4s12s23s34 − c13c24c34c3c4s12s14s2 + c34c3c4s12s13s23s24s2 +
+c12c3c4[−c13c24c34c2s14 + c34s24(c2s13s23 − c23s2) + s34(c23c2s13 +
+s23s2)] − c24c34c4s13s14s3 − c13c34c4s23s24s3 + c23c4[c34c2c3s12s24 +
+s34(c3s12s13s2 − c13s3)] + c14c24c34s4
+TABLE I: Ci-factors of the trilinear gauge couplings ZZhi and W ±W ∓hi as defined in
+Eq. (15) in the main text. Here cij = cos αij (sij = sin αij) and ci = cos βi (si = sin βi). In
+this notation, sij−k stands for sin(αij − βk).
+27
+
+Appendix B: General Yukawa Couplings
+General Yukawa Neutral Scalar
+ξud
+h
+c12c13c14 / c2c3c4
+ξe
+h
+s12c13c14 / s2c3c4
+ξµ
+h
+s13c14 / s3c4
+ξτ
+h
+s14 / s4
+ξud
+h2
+− (c23c24s12 + c12 (c24s13s23 + c13s14s24)) / c2c3c4
+ξe
+h2
+(c12c23c24 − s12 (c24s13s23 + c13s14s24)) / s2c3c4
+ξµ
+h2
+(c13c24s23 − s13s14s24) / s3c4
+ξτ
+h2
+c14s24 / s4
+ξud
+h3 (s12 (c34s23 + c23s24s34)−c12 (c13c24s14s34 + s13 (c23c34−s23s24s34))) /c2c3c4
+ξe
+h3 −(c12 (c34s23+c23s24s34)+s12 (c13c24s14s34+s13 (c23c34+s23s24s34)))/s2c3c4
+ξµ
+h3
+(−c24s13s14s34 + c13 (c23c34 − s23s24s34)) / s3c4
+ξτ
+h3
+c14c24s34 / s4
+ξud
+h4 (s12 (c23c34s24 − s23s34)−c12 (c13c24c34s14−s13 (c34s23s24 + c23s34))) /c2c3c4
+ξe
+h4 −(c12 (c23c34s24−s23s34)+s12 (c13c24c34s14−s13 (c34s23s24+c23s34)))/s2c3c4
+ξµ
+h4
+− (c24c34s13s14 + c13 (c34s23s24 + c23s34)) / s3c4
+ξτ
+h4
+c14c24c34 / s4
+TABLE II: General Yukawa couplings of the scalar Higgs particles to quarks and charged
+leptons, as defined in Eqs. (16) and (17) in the main text. Here cij = cos αij (sij = sin αij)
+and ci = cos βi (si = sin βi).
+28
+
+General Yukawa Pseudoscalar
+ξq
+A1
+− (c23c24s2 + c2 (c24s3s23 + c3s4s24)) / c2c3c4
+ξe
+A1
+(c2c23c24 − s2 (c24s3s23 + c3s4s24)) / s2c3c4
+ξµ
+A1
+(c3c24s23 − s3s4s24) / s3c4
+ξτ
+A1
+s24c4 / s4
+ξq
+A2
+(s2 (c34s23 + c23s24s34) − c2 (c3c24s4s34 + s3 (c23c34 − s23s24s34)))/c2c3c4
+ξe
+A2
+−(c2 (c34s23 + c23s24s34)+s2 (c3c24s4s34 + s3 (c23c34−s23s24s34)))/s2c3c4
+ξµ
+A2
+(−c24s3s4s34 + c3 (c23c34 − s23s24s34)) / s3c4
+ξτ
+A2
+c24s34c4 / s4
+ξq
+A3
+(s2 (c23c34s24 − s23s34) − c2 (c3c24c34s4 − s3 (c34s23s24 + c23s34)))/c2c3c4
+ξe
+A3
+−(c2 (c23c34s24 − s23s34)+s2 (c3c24c34s4−s3 (c34s23s24 + c23s34)))/s2c3c4
+ξµ
+A3
+− (c24c34s3s4 + c3 (c34s23s24 + c23s34)) / s3c4
+ξτ
+A3
+c24c34c4 / s4
+TABLE III: General Yukawa couplings of the pseudoscalar Higgs particles to quarks and
+charged leptons, as defined in Eqs. (16) and (17) in the main text. Here cij = cos γij
+(sij = sin γij) and ci = cos βi (si = sin βi).
+29
+
+General Yukawa Charged
+ξqLR
+H+
+1
+− (c23c24s2 + c2 (c24s3s23 + c3s4s24)) / c2c3c4
+ξeL
+H+
+1
+(c2c23c24 − s2 (c24s3s23 + c3s4s24)) / s2c3c4
+ξµL
+H+
+1
+(c3c24s23 − s3s4s24) / s3c4
+ξτL
+H+
+1
+s24c4 / s4
+ξqLR
+H+
+2
+(s2 (c34s23 + c23s24s34) − c2 (c3c24s4s34 + s3 (c23c34 − s23s24s34)))/c2c3c4
+ξeL
+H+
+2
+−(c2(c34s23 + c23s24s34)+s2 (c3c24s4s34 + s3 (c23c34 − s23s24s34)))/s2c3c4
+ξµL
+H+
+2
+(−c24s3s4s34 + c3 (c23c34 − s23s24s34)) / s3c4
+ξτL
+H+
+2
+c24s34c4 / s4
+ξqLR
+H+
+3
+(s2 (c23c34s24 − s23s34) − c2 (c3c24c34s4 − s3 (c34s23s24 + c23s34)))/c2c3c4
+ξeL
+H+
+3
+(c2 (s23s34 − c23c34s24) − s2 (c3c24c34s4 − s3 (c34s23s24 + c23s34)))/s2c3c4
+ξµL
+H+
+3
+− (c24c34s3s4 + c3 (c34s23s24 + c23s34)) / s3c4
+ξτL
+H+
+3
+c24c34c4 / s4
+TABLE IV: General Yukawa couplings of the charged Higgs particles to quarks and
+leptons, as defined in Eqs. (18) and (19) in the main text. Here cij = cos δij (sij = sin δij)
+and ci = cos βi (si = sin βi).
+30
+
+Appendix C: Benchmark Points
+Scalar benchmark points
+S1
+S2
+β2/π, β3/π, β4/π
+0.05, 0.16, 0.18
+0.04, 0.14, 0.21
+α23/π, α24/π, α34/π
+−0.09, −1.00, −0.70
+−0.02, −0.05, 0.10
+γ23/π, γ24/π, γ34/π
+0.50, 0.59, 0.80
+0.16, 0.52, 0.39
+δ23/π, δ24/π, δ34/π
+0.08, −0.26, −0.96
+0.62, −0.93, −0.95
+mh2, mh3, mh4 (GeV)
+269, 396, 483
+175, 359, 360
+mA1, mA2, mA3 (GeV)
+439, 454, 484
+265, 351, 369
+mH±
+1 , mH±
+2 , mH±
+3 (GeV)
+438, 441, 443
+289, 352, 370
+m2
+qe, m2
+qµ, m2
+qτ (GeV2)
+−17700, 71700, −340000 16000, −34600, −168000
+m2
+eµ, m2
+eτ, m2
+µτ (GeV2)
+−18600, 20700, −53600
+14000, −31200, −57400
+BR(h2 → ee)
+2.72 × 10−3
+1.63 × 10−4
+BR(h2 → µµ)
+4.68 × 10−1
+7.85 × 10−6
+BR(h2 → ττ)
+1.22 × 10−1
+7.42 × 10−1
+TABLE V: Benchmark points for the leptonic decays of the lightest neutral scalar (other
+than the Standard Model Higgs) from Figure 2, for a h2-mass range below 350 GeV.
+31
+
+Charged benchmark
+points
+C1
+C2
+β2/π, β3/π, β4/π
+0.05, 0.05, 0.09
+0.10, 0.16, 0.11
+α23/π, α24/π, α34/π
+0.09, 0.54, 0.34
+0.20, 0.88, 0.72
+γ23/π, γ24/π, γ34/π
+−0.04, 0.66, 0.60
+0.68, 0.50, −0.52
+δ23/π, δ24/π, δ34/π
+−0.98, 0.00, −0.36
+1.00, 0.00, 0.77
+mh2, mh3, mh4 (GeV)
+127, 187, 208
+180, 237, 240
+mA1, mA2, mA3 (GeV)
+131, 179, 244
+161, 172, 173
+mH±
+1 , mH±
+2 , mH±
+3 (GeV)
+164, 172, 229
+158, 181, 234
+m2
+qe, m2
+qµ, m2
+qτ (GeV2)
+−14800, −17400, 6210 57000, −127000, −15100
+m2
+eµ, m2
+eτ, m2
+µτ (GeV2)
+5880, 22100, 9060
+−75600, −9570, 81300
+BR(H±
+1 → e±νe)
+2.24 × 10−3
+1.68 × 10−2
+BR(H±
+1 → µ±νµ)
+5.36 × 10−1
+6.91 × 10−3
+BR(H±
+1 → τ ±ντ)
+4.55 × 10−1
+5.23 × 10−1
+TABLE VI: Benchmark points for the leptonic decays of the lightest charged scalar from
+Figure 3, for a H±
+1 -mass range below 180 GeV.
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+

AtAyT4oBgHgl3EQfq_mZ/content/tmp_files/2301.00553v1.pdf.txt

@@ -0,0 +1,1085 @@
+LIGHTWEIGHT IMAGE INPAINTING BY STRIPE WINDOW TRANSFORMER WITH
+JOINT ATTENTION TO CNN
+Bo-Wei Chen⋆
+Tsung-Jung Liu⋆
+Kuan-Hsien Liu†
+⋆Department of Electrical Engineering and Graduate Institute of Communication Engineering, National Chung Hsing University, Taiwan
+†Department of Computer Science and Information Engineering, National Taichung University of Science and Technology, Taiwan
+ABSTRACT
+Image inpainting is an important task in computer vision. As
+admirable methods are presented, the inpainted image is get-
+ting closer to reality. However, the result is still not good
+enough in the reconstructed texture and structure based on
+human vision. Although more and more larger models have
+been proposed recently because of the advancement of com-
+puter hardware, we would like to build a suitable model for
+personal use or small-sized institution. Therefore, we propose
+a lightweight model that combines the special transformer and
+the traditional convolutional neural network (CNN). Further-
+more, we noticed most researchers only consider three pri-
+mary colors (RGB) in inpainted images, but we think this
+is not enough so we propose a new loss function to inten-
+sify color details. Extensive experiments on commonly seen
+datasets (Places2 and CelebA) validate the efficacy of our pro-
+posed model compared with other state-of-the-art methods.
+Index Terms— HSV color space, image inpainting, joint
+attention mechanism, stripe window, vision transformer
+1. INTRODUCTION
+Image inpainting has been studied by many researchers for
+several years. The main goal of image inpainting is to fill
+up the realistic pixels in the missing region of the image and
+this can be applied to object removal and photo restoration.
+To achieve realistic results, we need to consider the follow-
+ing two important points: 1) the continuity of adjacent tex-
+tures; 2) visually reasonable structure. All the proposed meth-
+ods target at the above two points to solve the problem, such
+as the traditional diffusion method, patch matching method
+and current methods (CNN and GAN). However, they still
+face some difficulties because convolution-based CNN has
+a narrow receptive field and hence it cannot get the global
+information for the whole image. Without the global infor-
+mation of the whole image, it is hard to repair the key edge
+and lines of the scene. Some researchers proposed methods
+that utilize auxiliary information for structure recovery, e.g.,
+edges [1]. On the other hand, some researcher proposed an at-
+tention mechanism-based model using attention scores com-
+pared with each patch to get global information. Suvorov et
+al. [2] utilize the Fast Fourier Convolution (FFC) to encode
+features in the frequency domain with global receptive fields
+for resolution-robust inpainting. Although these methods im-
+prove the overall repair result but also causes a huge compu-
+tational cost. Furthermore, in recent years, the transformer
+has also been used in the inpainting field. It has the advantage
+of wider receptive fields than CNNs and better inpainting at
+low resolutions. Unfortunately, transformers require a lot of
+computer memory.
+Therefore, it inspires us to design a lightweight trans-
+former block with stable repair effects. Specifically, we use
+the CSWin transformer [3] which used stripe window self-
+attention to replace the traditional full self-attention. Stripe
+window self-attention mechanism computes self-attention
+parallel to horizontal and vertical stripe cross-windows. Each
+stripe is obtained by dividing the input feature into constant-
+width stripes. In this way, we can achieve global attention
+with limited computational cost and we redesign the trans-
+former block to improve the repair performance.
+The consistency of color is another important factor to
+judge the quality of the image. It is easy to discern the differ-
+ence between inpainted image and original image by human
+vision if the color has deviation. Most researchers only deal
+with basic primary colors but we think this is not enough. If
+we can quickly improve color consistency in the early stage of
+training, the repair performance can be improved. Therefore,
+we transform the inpainted image to HSV color space and
+compare it with the input image. In follow-up experiments,
+our method is confirmed to be effective.
+The rest of the paper is organized as follows. In Section
+2, we introduce the previous and state-of-the-art inpainting
+methods.
+Then we present our proposed method and loss
+function in Section 3. In Section 4, we exhibit our training
+details, experiment results, inpainting images, and ablation
+studies. At last, the conclusions are drawn in Section 5. Due
+to page limit, qualitative and quantitative results of CelebA
+dataset, object removal experiments, and other inpainting im-
+ages are provided in the Appendix (See Supplementary Ma-
+terial).
+arXiv:2301.00553v1  [eess.IV]  2 Jan 2023
+
+2. RELATED WORK
+Traditional inpainting.
+Traditional inpainting can gener-
+ally be divided into two categories. The first one is the dif-
+fusion method. Diffusion methods disseminate the texture
+content by one or multi-curve information from the known
+region to missing region.
+The second one is the patch-
+matching method.
+Patch matching [4] used approximate
+nearest-neighbor to find the nearest-neighbor region of the
+specified region and then selected the most similar nearest-
+neighbor region to fill in and complete image inpainting.
+The former method is easy to blur inpainting results in large
+masks, and the latter method will cost a lot of calculations.
+Deep learning based inpainting.
+With the advancement
+of hardware technology, CNN based deep learning model has
+become the mainstream. Gradually, more and more novel
+CNN models based on different modules have been proposed,
+e.g., some models utilize edge auxiliaries information, such as
+Nazeri et al. proposed Edgeconnect [1], Yu et al. proposed
+GateConv [5] which used Canny edge to generate edge im-
+ages. These methods used additional auxiliaries information
+to get more data to help the repair, which are really helpful
+in inpainting images with complex structure, such as build-
+ing and interior space, but inevitably they need more stages
+or parameters in training these methods. We also used edge
+information for the mask instead of the input in our proposed
+approach to enhance the edge structure.
+On the other hand, some researchers use contextual at-
+tention to enhance the texture inpainting, such as Yu et al.’s
+DeepFill [6], Zhu et al.’s MADF [7], Yi et al.’s HiFill [8].
+They calculate complicated attention scores to find the most
+similar texture that can be filled in the missing region. Gen-
+erally speaking, this type of methods is better than others in
+terms of texture. In our proposed method, we redesign the
+attention module and combine wide attention to the local re-
+ceptive field to achieve attention sharing.
+Vision transformer.
+Recently He et al. proposed a model
+named Vision transformer (VIT) [9]. Transformer has been
+long and widely used in the field of NLP. They made the trans-
+former usable in computer vision by their proposed method.
+As more novel transformers are proposed, e.g., Dong et al.’s
+CSWin transformer [3], some of them have been seen in the
+field of image inpainting, such as Zheng et al.’s TFill [10].
+For huge mask, transformer can inpaint plausible textures by
+their special attention. In addition, transformer has wider re-
+ceptive field than traditional convolution but also needs more
+computing costs than convolution. Therefore, we redesigned
+the basic transformer, and then used the stripe window to di-
+vide the feature map to reduce the amount of calculation and
+obtain a better repair effects.
+To summarize, this paper proposed a novel stripe-
+window-based special transformer framework for image in-
+painting, and enhanced it with joint attention local CNN lay-
+ers. Our model focuses on the global CSWin transformer and
+CNN-based local layer. We process the global and local layer
+in parallel and then share the same attention information be-
+tween them. In the end, we use four simple up-samples to get
+the inpainting result. The major contributions of this work are
+as follows:
+• We propose a stripe window self-attention transformer
+with an efficient local enhancement position encoding.
+Then we redesign the transformer block to make the
+result better than the original method.
+• We suggest joint attention from global layers to local
+layers, connecting the two layers to enhance the overall
+consistency of repair results.
+• We propose a new HSV loss focused on color consis-
+tency in the early stage.
+• In the common dataset including Places2, we conduct
+extensive experiments to confirm that our proposed
+model is better than other advanced methods.
+3. METHODOLOGY
+Overview.
+The whole model of our proposed approach is
+shown in Fig. 1. Given a masked image Im, and a binary
+mask M both in 256×256, we concatenate and input them
+to the three downsample CNN layers. After we downsam-
+ple input image, we split the channel to global layer (i.e.,
+CSWin transformer) and local residual in residual dense block
+(RRDB) [12] layer, where we use joint attention with differ-
+ent receptive fields between two layers. Each RDB block in
+RRDB has four consecutive Conv-ReLU. At last we concate-
+nate the features from both channels and then go through three
+upsample layers to get the inpainted image Iout.
+3.1. Special CSWin Transformer
+The overall global layer special CSWin Transformer is shown
+in Fig. 1. The input of the global layer is a feature map with
+size of H×W×C, where H and W are 32 after downsam-
+pling and the channel is 128 after the split. There are four
+CSWin transformer blocks in our global layer. Each block
+has its own multi-head and stripe window (sw) to reduce the
+amount of calculation. We set multi-head to 2, 4, 8, 16 and
+sw to 4, 8, 16, 32 for four blocks by default. The first three
+blocks are our special CSWin transformer block. They will
+split their channel into horizontal and vertical stripes, and
+then split their channel with their own multi-head again. The
+sw will split H or W chosen by horizontal stripes or vertical
+stripes. Different from the general multi-head self-attention
+(MHSA), our stripe window multi-head self-attention (SW-
+MHSA) combines multi-head and sw to greatly reduce the
+amount of calculation and achieve a better inpainting effect.
+After we get the split low-resolution image, we can do the
+
+Fig. 1: The overview of our proposed model. The whole model structure shows the framework of our proposed model and
+the details of the joint attention between Global layer and Local layer. The input images only include Im and M, and the Iedge
+will not be trained in the model and be generated by Canny [11] before training. Moreover, the right side shows the CSWin
+Transformer Block. D is the normalization factor before softmax, which makes the similarity between pixels become more
+stable. At last, the Residual Dense Block in the local layer is shown at the top right corner of the whole model.
+self-attention through Q(query), K(keys), V (values) until
+the last block. The last block of the CSWin transformer is the
+full attention because the sw in the fourth block is 32, which
+means the stripe window is the whole image.
+Redesigned CSWin Block.
+The structure of CSWin
+Block is also shown in Fig. 1. We redesign the self-attention
+wiring, moving it from the first feed-forward to the beginning
+because we hope our self-attention block will not be influ-
+enced by the SW-MHSA. Stripe Window Self-Attention and
+Full Self-Attention will be trained from different receptive
+fields and then connected together with the residual link. We
+also add locally-enhanced positional encoding (LePE) in the
+transformer block to augment the positional encoding and re-
+fer to [3] to add the LePE at the end of the transformer block
+but not the middle, shown on the right side of Fig. 1. We
+found that self-attention needs to be calculated multiple times
+to get better attention information. We set the Ni to denote
+the number of repetitions.
+3.2. Joint attention
+We concatenate global and local layers to jointly focus on
+the information with different receptive fields.
+We expect
+our inpainting results to be the admixture of different recep-
+tive fields, not only just global but also local receptive fields.
+So we collect attention from the second and fourth CSWin
+transformer blocks and multiply it by the corresponding RDB
+blocks. The dimensionality of RDB features is not the same
+as attention so we need to reshape the RDB feature to con-
+form to attention, like values in Q, K, V . At last two mixed
+receptive fields are added to the respective last block of the
+two layers to achieve joint attention.
+3.3. Loss Function
+Most loss functions we adopt are the same as [1,13,14]. And
+we also use other losses including Edge loss and HSV loss
+which we proposed in this work. First, the basic L1 func-
+tion is described as L1 = |Iout − IGT |, where Iout, IGT
+indicate predicted images and the ground truth, respectively.
+In addition to this, we also enhance the edge of the inpainting
+image by using Edge loss which is Ledge = 1
+n
+�n
+i=1 ||(Iout ⊙
+Medge − IGT ⊙ Medge)||2
+2. where n represents the number of
+pixels in the image, and Medge = (1 − Iedge) + 10 ∗ Iedge,
+which can be seen as an edge mask to accentuate the edge
+structure. The Iedge is the image obtained from Canny edge
+detection [11].
+In order to improve the quality of the inpainting model, we
+use Perceptual loss to measure the similarity between images.
+We also use the mask on feature map to let our Perceptual
+loss only focus on visible regions. The VGG-based perceptual
+loss would force the model to generate images semantically
+closer to the ground truth, but we notice our inpainting results
+have checkerboard artifacts. According to [14], checkerboard
+artifacts are usually caused by deconvolution and using Style
+loss can remove this artifact. Therefore, we use the same Style
+loss as [14] in our total loss.
+Besides focusing on texture and structure, we believe that
+color is as important as both. So we proposed the HSV loss to
+measure the similarity between colors, which can be formu-
+
+CSWin Transformer
+MLP2
+CNN
+CNN
+>SoftMax( -
+七.
+RRDB 
+★V
+>LePE(V)
+4
+Transformer bolck
+LN
+4
+MLP1
+CsWin
+CsWin
+CsWin
+CsWin
+ Tansformer
+Tansformer
+Tansformer
+Tansformer
+Block
+Block
+Block
+Block
+x Ni
+ x Ni
+ x Ni
+x Ni
+LN
+X
+Full
+Stripe Window
+ Self-Attention
+ Self-Attention
+RDB
+RDB
+RDB
+RDB
+Block
+Block
+Block
+Block
+LN
+LNlated as follows:
+LHSV = 1
+n
+n
+�
+i=1
+||(HSVout − HSVGT )||2
+2,
+LHSV edge = 1
+n
+n
+�
+i=1
+||(HSVout ⊙ Medge − HSVGT ⊙ Medge)||2
+2,
+LT otalHSV =λHSV ∗ LHSV + λHSV edge ∗ LHSV edge,
+(1)
+where λHSV = 10 and λHSV edge = 100 by default. Here,
+HSV means Hue, Saturation, V alue in HSV color space
+but we do not use V alue in the HSV loss because brightness
+(intensity) can easily be included by other losses. If we still
+use the V alue in HSV loss it will even affect our inpainting
+results. We demonstrated this in ablation experiments.
+The adversarial loss includes the discriminator loss LD
+and the generator loss LG. The adversarial loss can be indi-
+cated as
+LD = −EIGT [logD(IGT )] − EIoutM [logD(Iout) ⊙ (1 − M)]
+− EIoutM [log(1 − D(Iout)) ⊙ M],
+LG = −EIout[logD(Iout)],
+Ladv = LD + LG + λGP LGP ,
+(2)
+where the PatchGAN [15] based discriminator is written as D
+and our proposed model can be seen as the generator G. The
+LGP = EIGT || ▽IGT D(IGT )||2 is the gradient penalty and
+λGP = 1e − 3. We include all losses above as the total loss
+Ltotal:
+Ltotal = λL1L1 + λedgeLedge + λpercLperc
++ λstyleLstyle + λT otalHSV LT otalHSV + λadvLadv,
+(3)
+where λL1 = 10, λedge = 10, λperc = 0.1, λstyle = 250,
+λT otalHSV = 1, and λadv = 10. The above loss weights are
+empirically set by experiments.
+4. EXPERIMENTS
+4.1. Datasets
+To show the inpainting effectiveness of our proposed model,
+we conduct experiments on Places2 dataset. For Places2, we
+randomly chose 20k images from the original dataset as the
+training dataset, 5k images as the validation, and use about
+4k images as the test. we use less data and the lightweight
+model to show our proposed approach has better robustness
+than other state-of-the-art huge-parameters models. For all
+of the images in Places2 dataset, we only train and test them
+with image size 256×256. For other comparison methods, we
+use their provided pretrained model to perform the test on the
+same dataset as we did.
+4.2. Reference State-of-the-Art
+We compare the proposed model with other state-of-the-art
+methods, which include PatchMatch (PM) [4], Contextual At-
+tention (CA) [6], Shift-net (SN) [16], Partial Convolutions
+(PC) [14], Region-wise (RW) [17], Gated Convolution (Deep-
+Fill v2) [5], Contextual Residual Aggregation (HiFill) [8], Im-
+puted Convolution (Iconv) [18], Aggregated contextual trans-
+formations (AOT-GAN) [19], Mask-Aware Dynamic Filter-
+ing (MADF) [7], Auxiliary Contextual Reconstruction (CR-
+Fill) [20], Bridging Global Context Interactions (TFill) [10] ,
+Large Mask inpainting (LaMa) [2].
+4.3. Quantitative Comparisons
+In Table 1, we utilize PSNR and SSIM [21] to assess the
+performance of all compared methods and our proposed ap-
+proach on the Places2 dataset in image size 256×256 with
+irregular masks of different masking rates. The required pa-
+rameters are also shown below each method, where the re-
+sults are either tested by ourselves or can be referred to [2].
+For Places2, our proposed method can defeat most of com-
+pared methods in terms of these two evaluation metrics. On
+the other hand, our training images and steps are also less than
+most methods. Hence, our proposed model will surpass them
+if we have similar resources as they do.
+In Table 2, we utilize LPIPS [22] to assess the perceptual
+similarity of the compared methods on the Places2 dataset in
+image size 256×256 with irregular masks. We consider the
+LPIPS metric is more fair in the inpainting field because the
+main point of inpainting images is to reconstruct the image
+close to the real one. The perceptual similarity is more like
+what human vision sees. For Places2, our proposed model can
+achieve the best results among all compared methods. This
+means our inpainting images are closer to the real than other
+compared methods.
+4.4. Qualitative Comparisons
+We show the qualitative inpainting results of Places2 in Fig.
+2. Compared with other methods, our proposed model can
+reconstruct similar or even more clear textures. We notice our
+inpainting results are slightly blurred when we focus more on
+the transformer and less on CNN. In the future, we will set
+restrictions on the local layers so that local information will
+not be ignored. Furthermore, our architecture is a lightweight
+model, which means we do not need lots of parameters, but
+still can achieve similar results compared to those larger mod-
+els. Note that both our training data and steps are less than
+other methods.
+4.5. Ablation Study
+To confirm our proposed module and new loss function are
+useful in the proposed architecture, we separately test them
+in the ablation experiments.
+We test the stability of the
+CSWin transformer and the redesign in Table 3. We retrained
+the CSWin transformer without redesign and original trans-
+former [9] separately and compared them with our redesigned
+
+Table 1: Quantitative evaluation of inpainting on Places2 dataset. We report Peak signal-to-noise ratio (PSNR) and structural
+similarity (SSIM) metrics. The ▲ denotes larger, and ▼ denotes lesser of the parameters compared to our proposed model.
+(Bold means the 1st best; Underline means the 2nd best; Italics means the 3rd best)
+Places2
+PSNR ↑
+SSIM ↑
+Parameters
+x106
+mask
+5%
+≀
+10%
+10%
+≀
+20%
+20%
+≀
+30%
+30%
+≀
+40%
+40%
+≀
+50%
+50%
+≀
+60%
+5%
+≀
+10%
+10%
+≀
+20%
+20%
+≀
+30%
+30%
+≀
+40%
+40%
+≀
+50%
+50%
+≀
+60%
+PM [2009]
+-
+22.8734
+21.5227
+19.7799
+17.2039
+17.3965
+14.9213
+0.9365
+0.8939
+0.8816
+0.7497
+0.7276
+0.5939
+CA [2018]
+3 ▼
+30.6980
+26.5750
+26.3226
+22.6366
+21.8994
+20.3658
+0.9616
+0.9102
+0.9032
+0.8162
+0.7749
+0.7102
+SN [2018]
+55 ▲
+24.4305
+23.0565
+22.9565
+22.6845
+20.5982
+18.3062
+0.8934
+0.8680
+0.8415
+0.8067
+0.7076
+0.5874
+PC [2018]
+49 ▲
+25.5658
+23.4294
+23.4746
+24.2262
+23.2751
+22.6612
+0.8791
+0.8446
+0.8338
+0.8290
+0.8028
+0.7680
+RW [2019]
+47 ▲
+33.6373
+29.1710
+28.7519
+25.1838
+24.3569
+22.8062
+0.9677
+0.9222
+0.9158
+0.8386
+0.8018
+0.7431
+DeepFill v2 [2019]
+4 ▼
+32.7413
+28.3293
+27.0149
+24.1172
+23.3908
+21.7128
+0.9664
+0.9205
+0.9040
+0.8353
+0.7986
+0.7322
+HiFill [2020]
+3 ▼
+27.1280
+22.3913
+21.9062
+18.2817
+17.2410
+15.7043
+0.9302
+0.8254
+0.8043
+0.6713
+0.5796
+0.4878
+Iconv [2020]
+30 ▲
+27.6711
+23.6294
+23.1790
+20.3817
+19.3962
+18.3129
+0.9326
+0.8390
+0.8216
+0.7069
+0.6275
+0.5524
+AOT-GAN [2020]
+15 ▲
+31.0784
+28.2309
+27.9468
+24.5996
+23.7414
+22.1844
+0.9495
+0.9127
+0.9067
+0.8316
+0.7905
+0.7284
+CRFill [2021]
+4▼
+33.1914
+28.7165
+27.4195
+24.4297
+23.6835
+21.9153
+0.9684
+0.9223
+0.9107
+0.8421
+0.8033
+0.7277
+TFill [2022]
+15 ▲
+32.6788
+27.8063
+27.3391
+23.8051
+22.9376
+21.4181
+0.9642
+0.9141
+0.9062
+0.8277
+0.7873
+0.7293
+Ours [2023]
+6
+31.1749
+28.7178
+27.7527
+24.8424
+24.1266
+22.8663
+0.9443
+0.9232
+0.9117
+0.8493
+0.8036
+0.7343
+Fig. 2: Qualitative results of Places2 dataset among all compared models. From left to right: Masked image, RW [17], DeepFill
+v2 [5], HiFill [8], Iconv [18], AOT-GAN [19], CRFill [20], TFill [10], and Ours. Zoom-in for details.
+Table 2: Quantitative comparisons of Learned perceptual im-
+age patch similarity (LPIPS)
+LPIPS ↓
+RW [2019]
+DeepFill v2 [2019]
+HiFill [2020]
+Iconv [2020]
+AOT-GAN [2020]
+0.149
+0.155
+0.180
+0.161
+0.149
+MADF [2021]
+CRFill [2021]
+TFill [2022]
+LaMa [2022]
+Ours [2023]
+0.139
+0.1217
+0.1331
+0.135
+0.1156
+CSWin transformer. For the results shown in Table 3, our pro-
+posed approach has the best PSNR and SSIM.
+We also conduct experiments for HSV loss in Table 3.
+We noticed the Value (V) of HSV can easily be learned in L1
+and other losses. If we still consider V in LHSV , it will in-
+fluence the balance of the inpainting result, as shown in the
+table. We show the color deviation between with and with-
+out LT otalHSV in early training steps in Fig.
+3.
+We can
+see the color of the inpainting results in the early 50 train-
+ing steps, which shows the one with LT otalHSV is more close
+to the ground truth than without LT otalHSV , and the known
+region and the missing region are more consistent when using
+Table 3: Ablation study of HSV loss and redesigned special
+CSWin transformer with size 256×256 images on Places2
+PSNR↑
+SSIM ↑
+LPIPS↓
+original transformer
+25.7935
+0.8072
+0.1242
+w/o redesigned CSWin
+26.1027
+0.8377
+0.1221
+ours w/o HSV loss
+26.2786
+0.8459
+0.1212
+ours w/ full HSV loss
+26.4757
+0.8541
+0.1184
+ours w/ redesigned CSWin
+and HSV loss (w/o V)
+26.5801
+0.8611
+0.1156
+Fig. 3: Ablation study of color deviation on inpainted images.
+From left to right: Masked images, w/o TotalHSV loss, and
+TotalHSV loss (w/o V).
+
+LT otalHSV .
+5. CONCLUSION
+In this paper, we propose a lightweight joint attention trans-
+former architecture. We use transformer-based architecture to
+get wide receptive field information and cooperate with local
+layers with RRDB by joint attention with each other. Our pro-
+posed HSV loss can stabilize the colors in early training steps
+and eventually further improve the inpainting performance.
+We use the CSWin transformer and redesign the transformer
+block to not confuse the two self-attentions and achieve sig-
+nificant improvements. Our experiments demonstrate that our
+proposed model using small amount of parameters can still
+generate similar or even better inpainting results than other
+state-of-the-art methods. Those large models do have an ad-
+vantage in details but not every researcher has enough hard-
+ware support. Therefore we propose this approach to demon-
+strate small models are also able to compete with large mod-
+els.
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+1134, 2017.
+[16] Z. Yan, X. Li, M. Li, W. Zuo, and S. Shan, “Shift-net:
+Image inpainting via deep feature rearrangement,” in
+Proceedings of the European conference on computer
+vision (ECCV), pp. 1–17, 2018.
+[17] Y. Ma, X. Liu, S. Bai, L. Wang, A. Liu, D. Tao, and
+E. R. Hancock, “Regionwise generative adversarial im-
+age inpainting for large missing areas,” IEEE Transac-
+tions on Cybernetics, 2022.
+[18] H. Hukkel˚as, F. Lindseth, and R. Mester, “Image in-
+painting with learnable feature imputation,” in DAGM
+German Conference on Pattern Recognition, pp. 388–
+403, Springer, 2020.
+[19] Y. Zeng, J. Fu, H. Chao, and B. Guo, “Aggregated
+contextual transformations for high-resolution image
+inpainting,” IEEE Transactions on Visualization and
+Computer Graphics, 2022.
+[20] Y. Zeng, Z. Lin, H. Lu, and V. M. Patel, “Cr-fill: Gen-
+erative image inpainting with auxiliary contextual re-
+construction,” in Proceedings of the IEEE/CVF Inter-
+national Conference on Computer Vision, pp. 14164–
+14173, 2021.
+[21] Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simon-
+celli, “Image quality assessment: from error visibility
+to structural similarity,” IEEE transactions on image
+processing, vol. 13, no. 4, pp. 600–612, 2004.
+[22] R. Zhang, P. Isola, A. A. Efros, E. Shechtman, and
+O. Wang, “The unreasonable effectiveness of deep
+features as a perceptual metric,” in Proceedings of
+
+the IEEE conference on computer vision and pattern
+recognition, pp. 586–595, 2018.
+7. APPENDIX
+7.1. Qualitative and Quantitative Results in CelebA
+Dataset
+Datasets For the CelebA, we use the whole dataset of
+CelebA and split them with the ratio of 8:1:1 for the train,
+validation, and test datasets.
+For all of the images in
+CelebA dataset, we only train and test them with image
+size 256×256. For other comparison methods, we use their
+provided pretrained model to perform the test on the same
+dataset as we did.
+Reference State-of-the-Art We compare the proposed
+model with other state-of-the-art methods, which include
+PatchMatch (PM), Contextual Attention (CA) , Shift-net
+(SN), Partial Convolutions (PC), Region-wise (RW), Gated
+Convolution (DeepFill v2), Imputed Convolution (Iconv),
+Aggregated contextual transformations (AOT-GAN), Aux-
+iliary Contextual Reconstruction (CRFill), Bridging Global
+Context Interactions (TFill).
+Result We show the qualitative comparison in Table 4.
+We utilize PSNR and SSIM to assess the performance of
+all compared methods (including our proposed approach)
+on the CelebA dataset in image size 256×256 with irregu-
+lar masks of different masking rates. The required param-
+eters are also shown below each method, where the results
+are tested by ourselves. Although our proposed method lose
+slightly in tiny masks (5% to 10%), we can defeat most of
+compared methods in huge masks. On the other hand, our
+training steps are also less than most methods. Hence, our
+proposed model will surpass them if we have similar re-
+sources as they do. We show the qualitative inpainting re-
+sults of CelebA in Fig. 5. For CelebA, our inpainted results
+are slightly different from the ground-truth image. This hap-
+pens with too much focusing on global information and no
+limitation on local information filling.
+7.2. Other inpainting images
+We also exhibit more inpainting results in Fig. 6 (Places2)
+and Fig. 7 (CelebA). From top to bottom is small masks to
+the huge masks. Zoom-in for details.
+7.3. Object Removal
+We additionally conduct object removal experiments in Fig.
+4.
+Our proposed method did well in target removal and
+background repair. If the background is relatively single, the
+result will be better than the grid background. This means
+our model needs to enhance structure in inpainting images,
+which will be our future research. Our codes are released
+in https://github.com/bobo0303/LIGHTWEIGHT-IMAGE-
+INPAINTING-BY-STRIPE-WINDOW-TRANSFORMER-
+WITH-JOINT-ATTENTION-TO-CNN.
+Fig. 4: Object removal (size 256×256) results. From left to
+right: Original image, mask, object removal result.
+
+3Fig. 5: Inpainting (size 256×256) results of all compared models in the CelebA dataset. From left to right: Masked image, RW,
+DeepFill v2, Iconv, AOT-GAN, CRFill, TFill, and Ours. Zoom-in for details.
+Table 4: Quantitative evaluation of inpainting on CelebA dataset. We report Peak signal-to-noise ratio (PSNR) and structural
+similarity (SSIM) metrics. The ▲ denotes larger, and ▼ denotes lesser of the parameters compared to our proposed model.
+(Bold means the 1st best; Underline means the 2nd best; Italics means the 3rd best)
+CelebA
+PSNR ↑
+SSIM ↑
+Parameters
+x106
+mask
+5%
+≀
+10%
+10%
+≀
+20%
+20%
+≀
+30%
+30%
+≀
+40%
+40%
+≀
+50%
+50%
+≀
+60%
+5%
+≀
+10%
+10%
+≀
+20%
+20%
+≀
+30%
+30%
+≀
+40%
+40%
+≀
+50%
+50%
+≀
+60%
+PM [2009]
+-
+21.4397
+21.4637
+20.5820
+18.3917
+17.5311
+14.1646
+0.9280
+0.9094
+0.8693
+0.8174
+0.7731
+0.6605
+CA [2018]
+3 ▼
+34.5586
+29.5535
+29.2139
+25.1065
+24.3168
+22.4536
+0.9551
+0.9277
+0.9214
+0.8223
+0.8114
+0.7602
+SN [2018]
+55 ▲
+20.7527
+19.3196
+18.7572
+17.1757
+15.7176
+15.4753
+0.8222
+0.8181
+0.7620
+0.6727
+0.5793
+0.5373
+PC [2018]
+49 ▲
+24.9021
+23.2179
+23.3923
+22.3591
+21.0050
+22.4939
+0.8592
+0.8463
+0.8442
+0.8109
+0.7653
+0.7933
+RW [2019]
+47 ▲
+33.0057
+28.5174
+28.1346
+24.7360
+23.9071
+22.4122
+0.9612
+0.9045
+0.8947
+0.8177
+0.7695
+0.7140
+DeepFill v2 [2019]
+4 ▼
+33.2820
+28.6673
+28.6339
+25.1281
+24.5153
+22.5632
+0.9723
+0.9239
+0.8286
+0.8651
+0.8147
+0.7757
+Iconv [2020]
+30 ▲
+27.1739
+27.1739
+26.7287
+23.7124
+22.8412
+21.4760
+0.8774
+0.8774
+0.8629
+0.7820
+0.7192
+0.6660
+AOT-GAN [2020]
+15 ▲
+30.9702
+28.5580
+28.3887
+25.1810
+24.5387
+22.8271
+0.9461
+0.9145
+0.9094
+0.8539
+0.8214
+0.7722
+CRFill [2021]
+4▼
+35.1434
+29.2685
+28.6638
+25.6514
+24.5170
+22.8599
+0.9745
+0.9293
+0.9148
+0.8628
+0.8160
+0.7751
+TFill [2022]
+15▲
+32.5255
+27.4428
+27.0993
+23.0947
+22.3084
+20.5222
+0.9662
+0.9156
+0.9075
+0.8316
+0.7906
+0.7333
+Ours [2023]
+6
+31.7816
+28.8494
+28.7076
+25.9073
+24.6157
+22.9164
+0.9466
+0.9240
+0.9226
+0.8713
+0.8222
+0.7816
+
+Fig. 6: Other inpainting (size 256×256) results in the Places2 dataset. From left to right: Masked image, RW, DeepFill v2,
+HiFill, Iconv, AOT-GAN, CRFill, TFill, and Ours. From top to bottom is 5% mask to 60% mask. Zoom-in for details.
+
+Fig. 7: Other inpainting (size 256×256) results in the CelebA dataset. From left to right: Masked image, RW, DeepFill v2,
+Iconv, AOT-GAN, CRFill, TFill, and Ours. From top to bottom is 5% mask to 60% mask. Zoom-in for details.
+
+nupre
+50700光

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@@ -0,0 +1,1729 @@
+ENERGY DISTRIBUTION FOR DIRICHLET EIGENFUNCTIONS
+ON RIGHT TRIANGLES
+HANS CHRISTIANSON AND DANIEL PEZZI
+Abstract. In this paper, we continue the study of eigenfunctions on trian-
+gles initiated by the first author in [Chr17] and [Chr19]. The Neumann data
+of Dirichlet eigenfunctions on triangles enjoys an equidistribution law, being
+equidistributed on each side. The proof of this result is remarkably simple,
+using only the radial vector field and a Rellich type integrations by parts. The
+equidistribution law, including on higher dimensional simplices, agrees with
+what Quantum Ergodic Restriction would predict. However, distribution of
+the Neumann data on subsets of a side is not well understood, and elementary
+methods do not appear to give enough information to draw conclusions.
+In the present note, we first show that an “obvious” conjecture fails even
+for the simplest right isosceles triangle using only Fourier series. We then use
+a result of Marklof-Rudnick [MR12] in which the authors show an interior
+spatial equidistribution law for a density-one subsequence of eigenfunctions
+to give an estimate on energy distribution of eigenfunctions on the interior.
+Finally we present some numerical computations suggesting the behaviour of
+eigenfunctions on almost isosceles triangles is quite complicated.
+1. Introduction
+Eigenfunctions on bounded Euclidean domains are used to model many physical
+and mechanical phenomena, and can be used to construct solutions to separable
+partial differential equations such as wave and Schr¨odinger type equations. The
+study of eigenfunctions and solutions to wave type equations are so closely related
+that concepts like propagation and flow are often used to understand eigenfunctions.
+Since waves propagate along straight lines, it is reasonable to expect eigenfunctions
+to live along straight lines. Since waves meeting a smooth boundary transversally
+reflect according to Snell’s law, eigenfunctions have incoming and outgoing com-
+ponents as well.
+And since waves are very complicated near corners and other
+discontinuities, so are eigenfunctions.
+In this note, we study the distribution of interior energy of eigenfunctions on right
+triangles, which is a measure of phase space concentration. If the billiard flow on a
+reasonable domain is ergodic, then quantum ergodicity [Shn74,Zel87,CdV85,ZZ96]
+implies that a density one subsequence of the eigenfunctions equidistributes in both
+space and phase space. This paper investigates similar properties on triangles, how-
+ever no assumption about classical ergodicity is made, and instead the theoretical
+components of this paper rely on several previous results concerning the distribu-
+tion of Neumann data mass on sides proved by the first author in [Chr17] as well as
+the spatial equidistribution result of Marklof-Rudnick [MR12] on rational polygons.
+There are several results in this paper. First, we use integrations by parts to
+connect interior energy to certain weighted boundary integrals. A comparison with
+1
+arXiv:2301.03555v1  [math.AP]  9 Jan 2023
+
+2
+H. CHRISTIANSON AND D. PEZZI
+the results in [Chr17] suggests the eigenfunctions have nice phase space distribu-
+tion properties, however this appears to be false. In fact, even on the right isosceles
+triangle, the weighted boundary integrals are subtle, even though the phase space
+distribution follows from symmetry. Second, using the spatial distribution for ratio-
+nal polygons in [MR12], we prove that on rational right triangles there is a density
+one subsequence that has frequency localization estimates, but an asymptotic is
+unclear.
+Finally, we provide some numerical data which suggests the weighted
+boundary integrals do not have an asymptotic, or at least not for the whole se-
+quence of eigenfunctions.
+2. Theoretical results on right triangles
+Our first set of results is on the right isosceles triangle.
+Theorem 1. Let T be the right isosceles triangle in the xy-plane, oriented as
+T = {0 ≤ x ≤ 1, 0 ≤ y ≤ 1 − x}.
+Consider the Dirichlet eigenfunction problem on T:
+(1)
+�
+�
+�
+�
+�
+−h2∆u = u, on T,
+u|∂T = 0,
+∥u∥L2(T ) = 1,
+where ∆ = ∂2
+x + ∂2
+y and h−2 denotes the eigenvalues of −∆, taking discrete values.
+Then there is an orthonormal basis for L2 of eigenfunctions satisfying
+∥h∂xu∥2
+L2(T ) = ∥h∂yu∥2
+L2(T ) = 1/2.
+On the other hand, there exists a subsequence of these eigenfunctions whose Neu-
+mann data satisfies
+lim inf
+h→0
+�
+0≤x≤1/2
+|h∂νu(x, 0)|2dx > lim sup
+h→0
+�
+1/2≤x≤1
+|h∂νu(x, 0)|2dx.
+Remark 2. This theorem shows that this sequence of eigenfunctions has a subse-
+quence which is not quantum ergodic on the boundary, even though the eigenfunc-
+tions are weakly equidistributed in phase space.
+Our second main result is about phase space distribution on rational right trian-
+gles. The result is heavily dependent on the choice of orientation for the triangle,
+and rotating the triangle changes the result. This result is then meant merely as
+an example of what one can prove with elementary techniques.
+Let Ω ⊂ R2 be a right triangle, and consider the semiclassical Laplace eigenfunc-
+tion problem (1).
+After rescaling and rotating, assume Ω is oriented so that it can be written as
+Ω = {(x, y) : 0 ≤ x ≤ a, 0 ≤ y ≤ 1 − x
+a}. Let F1, F2, F3 be the sides as in Figure 1.
+We further assume that Ω is rational, meaning that all angles are rational mul-
+tiples of π. A result of Marklof-Rudnick [MR12] shows that in this case, there is a
+density one subsequence of eigenfunctions uj such that
+�
+U
+|uj|2dV → Area(U)
+Area(Ω)
+as h → 0. We will work with this subsequence, and prove the following Theorem:
+
+ENERGY DISTRIBUTION
+3
+F1
+F2
+F3 = {y = 1 − x
+a}
+0
+a
+1
+Figure 1.
+Setup for right triangles
+Theorem 3. Suppose Ω is the right triangle oriented as in Figure 1. Let {uj} be
+the sequence of orthonormal Dirichlet eigenfunctions on Ω. There exists a density
+one subsequence {ujk} such that
+lim sup
+k→∞
+�
+Ω
+|h∂xujk|2dV ≤ 7
+8.
+Remark 4. We again emphasize that this estimate is highly dependent on the
+orientation of the triangle. The same proof works for the y-derivatives, so that,
+given ϵ > 0, there exists K such that k ≥ K implies
+1
+8 − ϵ ≤
+�
+Ω
+|h∂xujk|2dV ≤ 7
+8 + ϵ.
+A rotation of the triangle into different coordinates (s, t) is
+h∂x = αh∂s + βh∂t, h∂y = −βh∂s + αh∂t,
+where α2 + β2 = 1. Plugging in to our estimate gives
+1
+8 − ϵ ≤
+�
+Ω
+|(αh∂s + βh∂t)ujk|2dV ≤ 7
+8 + ϵ
+and similarly
+1
+8 − ϵ ≤
+�
+Ω
+|(−βh∂s + αh∂t)ujk|2dV ≤ 7
+8 + ϵ.
+Expanding these quantities does not give us much information unless one of α or
+β is close to zero.
+Remark 5. The main idea of the proof is to estimate the mass of h∂xu in strips
+to that of u in strips. We then use this and the results from [Chr17] on Neumann
+data on a whole side to get weak estimates on partial Neumann data.
+2.1. Quantum Ergodicity. Roughly speaking, quantum ergodicity (QE) for pla-
+nar domains states that if the classical billiard flow is ergodic, then there is a
+density one subsequence of eigenfunctions which equidistribute in phase space
+[Shn74,Zel87,CdV85,ZZ96]. That is, this subsequence of eigenfunctions distributes
+evenly both on the domain and in frequency. The work of Lindenstrauss [Lin06]
+shows that quantum ergodicity can hold for the whole sequence of eigenfunctions,
+called quantum unique ergodicity (QUE). The work of Hassell [Has10] shows that
+QUE can fail, so the question of QUE versus non-QUE is very subtle.
+
+4
+H. CHRISTIANSON AND D. PEZZI
+In related work, Hassell-Zelditch [HZ04] show that the boundary Neumann
+data of Dirichlet (and Dirichlet data of Neumann) eigenfunctions satisfy a nat-
+ural quantum ergodic property, called quantum ergodicity of restrictions (QER).
+Work of Toth-Zeldtich [TZ12,TZ13] extend these results to interior hypersurfaces,
+again along a density one subsequence. The work of the first author and Toth-
+Zelditch [CTZ12] proves that QUE implies quantum unique ergodicity for restric-
+tions (QUER) to interior hypersurfaces, at the expense of needing both the (weighted)
+Dirichlet and Neumann data for the equidistribution.
+In [Chr17] (see also [Chr19] in higher dimensions), the first author proves that
+for any planar triangle, the Neumann data of Dirichlet eigenfunctions satisfies an
+equidistribution identity on each side:
+Theorem 6 ( [Chr17, Chr19]). Let T be a planar triangle with sides A, B, C of
+length a, b, c respectively.
+Consider the (semi-classical) Dirichlet eigenfunction
+problem (1) and assume the eigenfunctions are normalized (||u||2
+L2(T ) = 1).
+Then the (semi-classical) Neumann data on the boundary satisfies
+�
+A
+|h∂νu|2dS =
+a
+Area(T)
+(2)
+�
+B
+|h∂νu|2dS =
+b
+Area(T)
+(3)
+�
+C
+|h∂νu|2dS =
+c
+Area(T)
+(4)
+where h∂ν is the semi-classical normal derivative on ∂T, dS is the arc-length mea-
+sure, and Area(T) is the area of the triangle T.
+Remark 7. This property is called ‘equidistribution’ as the Neumann data on each
+side is proportional to the length of that side, and the quantities are exactly what
+would be predicted if QUER was satisfied on the boundary. However, we stress that
+the integrals need to be over the whole side. Distribution of Neumann data over
+subsets of the sides is the topic of this paper, and indeed Theorem 1 shows this fails
+in the simplest possible case of a right isosceles triangle.
+There are several natural questions that arise based on this result. What can be
+said about the Neumann data on subsets of sides? Can we get an analogous result
+for subsets, even if we only consider results in a high energy limit or subsequences
+of a specific density? What about volume integrals over the same domain?
+To answer these questions in Euclidean space, we will begin by dealing with the
+case of a right isosceles triangle as we have explicit solutions to work with. We will
+then move on to numerical results which will allow us to get data from triangles to
+properly set expectations for these tough analytical problems.
+2.2. Immediate Questions. Based on this result, this paper is concerned with
+two immediate questions.
+Question 1. Is it true that
+(5)
+∀ω ⊂ ∂T, lim
+h→0
+�
+ω
+|h∂νu|2dS →
+m(ω)
+Area(T),
+where m(ω) is the measure of the set ω?
+
+ENERGY DISTRIBUTION
+5
+This is just an extension of the equidistribution result to arbitrary subsets. A
+second obvious question would be the following:
+Question 2. Is it true that
+(6)
+∀h > 0,
+�
+T
+|h∂yu|2dV =
+�
+T
+|h∂xu|2dV = 1
+2?
+2.3. Connecting boundary integrals to interior energy. Let us continue to
+work with the right triangle given by Figure 1. We duplicate the argument from
+[Chr17] but with the vector field X = x∂x. The point is that X = 0 on {x = 0}
+and X is tangential on {y = 0}. Along the side F0 = {0 ≤ x ≤ a, 0 ≤ y ≤ 1−x/a}
+we have the tangent derivative is ∂τ = γ−1(a∂x − ∂y) where γ = (1 + a2)1/2. The
+normal derivative is then ∂ν = γ−1(∂x + a∂y). Since u = 0 along F0, we have
+∂τu = γ−1(a∂x − ∂y)u = 0, or ∂yu = a∂xu on F0. Hence
+∂νu = γ−1(∂x + a∂y)u = γ−1(1 + a2)∂xu = γ∂xu,
+so that
+∂xu = γ−1∂νu
+along F0.
+Then using the same integrations by parts as in [Chr17], we have
+�
+Ω
+([−h2∆ − 1, X]u)¯udV = 2
+�
+Ω
+(−h2∂2
+xu)¯udV = 2
+�
+Ω
+|h∂xu|2dV.
+On the other hand, unpacking the commutator and applying Green’s formula just
+like in [Chr17], we have
+�
+Ω
+([−h2∆ − 1, X]u)¯udV
+(7)
+=
+�
+∂Ω
+(hXu)h∂ν ¯udS
+(8)
+=
+�
+F0
+x(h∂xu)h∂ν ¯udS
+(9)
+= γ−1
+�
+F0
+x|h∂νu|2dS.
+(10)
+This shows that, if we knew that the Neumann data along F0 was equidistributed
+on subsets of the side, we would have
+γ−1
+�
+F0
+x|h∂νu|2dS = 1
+2γ−1
+�
+F0
+|h∂νu|2dS.
+From [Chr17] we know the integral on the right is equal to 1. Rearranging, this
+computation would tell us that
+�
+Ω
+|h∂xu|2dV = 1/2,
+however this “obvious” conjecture appears to be false.
+Similar computations with vector fields like X = y∂x connects the quantity
+�
+Ω(h∂xu)(h∂y¯u)dV to other weighted boundary integrals, so weighted boundary
+integrals are essential to understanding interior energy distribution of eigenfunc-
+tions.
+
+6
+H. CHRISTIANSON AND D. PEZZI
+3. Analytical Results for Right Isosceles Triangle
+3.1. Introducing the eigenfunctions on the Right Isosceles Triangle. As
+we have explicit formulas for eigenfunctions of the Laplacian on a right isosceles
+triangle, we will study these functions both to prove conclusively some results and as
+a baseline for results we discuss later on almost isosceles triangles. For this section,
+T will be a triangle in Euclidean space with vertices (0, 0), (1, 0), and (0, 1). For
+the rest of this paper, we will deal with triangles with vertices at the origin and at
+(0, 1). We will identify triangles by the x coordinate of the third vertex, which will
+always be on the positive x-axis.
+Theorem 8. Let T be as previously described. Then the following formula exhausts
+all of the eigenfunctions of the Laplacian on T that satisfy Dirichlet boundary con-
+ditions with m, n ∈ Z, m ̸= n.
+(11)
+umn = cmn sin(nπx) sin(mπy) + dmn sin(mπx) sin(nπy).
+With the additional constraint that cmn = dmn if m and n are of opposite parity
+and cmn = −dmn if m and n have the same parity. Additionally, by normalization,
+c2
+mn = d2
+mn = 4.
+Proof. We will show that these functions are exhaustive, satisfy the boundary con-
+ditions, and satisfy the eigenfunction equation.
+We achieve this expression by
+noticing that reflecting T across the line y = 1 − x gives a square. The eigenfunc-
+tions of the Laplacian on a square are well known, so we know immediately that
+this list is exhaustive. We then just have to check all of the usual requirements to
+verify these are indeed eigenfunctions on the isosceles triangle.
+Clearly x = 0 =⇒ umn = 0 and y = 0 =⇒ umn = 0. Checking y = 1 − x gives
+the following expression:
+umn(x, 1 − x) = cmn sin(nπx) sin(mπ − mπx) + dmn sin(mπx) sin(nπ − nπx)
+(12)
+= (−1)m+1cmn sin(nπx) sin(mπx) + (−1)n+1dmn sin(mπx) sin(nπx)
+(13)
+That umn solves the eigenfunction equation carries over from the fact that these
+are restricted eigenfunctions of the square. A simple computation gives the eigen-
+value as h−2 = π2(n2 + m2) which is the same as in the square case.
+□
+3.2. Calculating the Volume Integral for the Right Isosceles Triangle.
+One of the metrics we are interested in is
+�
+T |h∂yumn|2dV . We will refer to this as
+the “y volume integral” for expository convenience. The derivative integrals and
+the function integrals are related by the equation
+�
+T |h∂xumn|2 + |h∂yumn|2dV =
+�
+T |umn|2dV = 1. This expression can be achieved by simple integration by parts,
+as we have
+(14) h2
+�
+T
+∂xumn∂xumn+∂yumn∂yumndV =
+�
+T
+umn(−h2∆umn)dV =
+�
+T
+u2
+mndV,
+where the boundary terms are zero as we assume Dirichlet boundary conditions
+and the last substitution uses umn being an eigenfunction.
+As quantum ergodicity can be interpreted as most of the eigenfunctions tending
+towards equidistribution, and a consequence of this is the volume integrals of the
+
+ENERGY DISTRIBUTION
+7
+derivatives tending both tending to
+1
+2.
+In the simple case of the right isosceles
+triangle, we have equality in L2 norms of ∂xu and ∂yu by symmetry, so we in
+fact have equality for every eigenvalue.
+As the volume metrics are completely
+understood in this case, it is natural to investigate the analogous metrics on the
+boundary as well.
+Later on, the y volume integral will be an important metric throughout this
+paper as a way to test quantum ergodicity. We can calculate an integral over the
+entire domain to test if our functions are quantum ergodic compliant, which is far
+easier numerically than dealing with subsets of the domain.
+3.3. Showing Equidistribution fails for subsets of the boundary of the
+Right Isosceles Triangle. To begin addressing the question of what happens on
+subsets of sides, we will explore the amount of the Neumann data on one half of the
+side on the x-axis compared to the other. As the sum of the data on both halves of
+the bottom side is constant, we only consider the bottom left face. This calculation
+is the same for the left face, as we have xy symmetry. As such we will define:
+(15)
+Il(m, n) = 1
+2
+� 1/2
+0
+|h∂νumn(x, 0)|2dx
+(16)
+Ir(m, n) = 1
+2
+� 1
+1/2
+|h∂νumn(x, 0)|2dx.
+We always have Il(m, n) + Ir(m, n) = 1 by the result for Neumann data on the
+entire side in [Chr17]. Il = Ir = 1
+2 represents equidistribution on the two halves.
+This would not be enough to say the Neumann data is uniformly distributed, but
+we will see almost immediately that equidistribution does not hold even in this
+simple case.
+Theorem 9. There exists m, n such that Il(m, n) ̸= 1
+2. Moreover, the subsequence
+Il(k, k + 1) converges to a value other than 1/2.
+Proof. On the bottom side the normal derivative is just −∂y. By direct calculation:
+���h∂νu|y=0
+���
+2
+= h2�
+c2
+mnm2π2 sin2(nπx) + 2cmndmnπ2nm sin(nπx) sin(mπx)
++ d2
+mnn2π2 sin2(mπx)
+�
+,
+which has the following anti-derivative F (m ̸= n) using basic trig identities:
+F(x) = h2c2
+mnm2π2�1
+2x −
+1
+4nπ sin(2nπx)
+�
++ h2cmndmnπ2nm
+�
+1
+π(n − m) sin(π(n − m)x) −
+1
+π(n + m) sin(π(n + m)x)
+�
++ h2d2
+mnn2π2�1
+2x −
+1
+4mπ sin(2mπx)
+�
++ C
+This lets us calculate explicitly:
+
+8
+H. CHRISTIANSON AND D. PEZZI
+F(0)
+= 0,
+F(1/2)
+= h2π2(c2
+mnm2 + d2
+mnn2)
+4
++ h2cmndmnπnm
+�sin( 1
+2π(n − m)
+(n − m)
+− sin( 1
+2π(n + m))
+(n + m)
+�
+= 1 + h2cmndmnπnm
+�sin( 1
+2π(n − m)
+(n − m)
+− sin( 1
+2π(n + m))
+(n + m)
+�
+, and
+F(1)
+= h2
+2 (c2
+mnm2π2 + d2
+mnn2π2) = 2.
+This gives us the following:
+2Il(m, n) = F(1/2) − F(0)
+= 1 + h2cmndmnπnm
+�sin( 1
+2π(n − m))
+(n − m)
+− sin( 1
+2π(n + m))
+(n + m)
+�
+.
+Note that if m + n is even, then 2Il(m, n) = 1. We will then consider situations
+where m + n is odd, which forces cmndmn = 4 as m + n is odd when m and n have
+different parities. Furthermore, as we push m and n to infinity, the term multiplied
+by
+1
+n+m will go to 0 as h2 = (π2n2 + π2m2)−1. We will numerically show what
+all of these values are later on, but to construct our subsequence consider mk = k
+and nk = k + 1. This is a subsequence for which the terms multiplied by
+1
+n−m will
+have the largest magnitude. Restricting to this subsequence and plugging in exact
+values for h2cmndmn gives us:
+2I1(mk, nk) = 1 + 4(π2(2k2 + 2k + 1))−1π(k2 + k)
+�
+sin(π
+2 ) − sin( π
+2 (2k + 1))
+2k + 1
+�
+∼ 1 + 2
+π + O(k−1)
+This implies that, for this subsequence of proportions Il(k, k + 1), we have that
+Il(k, k + 1) → 1
+2 + 1
+π ≈ .8183. It is then immediately the case that Ir(k, k + 1) →≈
+.1817.
+This is our subsequence that does not equidistribute on subsets in the
+limit.
+□
+The lack of equidistribution on subsets of the sides, even in the limit, is more sur-
+prising than the y volume integral result. This contradicts the original conjecture
+that there was a uniform distribution in the limit. In this case the long term behav-
+ior of these proportions can be completely described. The following computation
+is identical to the previous one but done in generality.
+Corollary 10. Let m and n be integers such that n − m = j where j is an odd
+integer. Then we have an explicit formula for Il(m, n).
+
+ENERGY DISTRIBUTION
+9
+Proof. We proceed in the same manner as the previous proof. By plugging in our
+assumed values we have:
+2Il(m, m + j) = 1 + 4π−1(2m2 + 2mj + j2)−1(m2 + mj)(sin( π
+2 j)
+j
+− sin( π
+2 (2m + j))
+2m + j
+)
+∼ 1 + δ(j) 2
+jπ + O(m−1)
+and therefore
+Il(m, n) ∼ 1
+2(1 + δ(j) 2
+jπ + O(k−1))
+Ir(m, n) ∼ 1
+2(1 − δ(j) 2
+jπ + O(k−1)),
+where δ(j) = 1 if j ≡ 1 (mod 4) and δ(j) = −1 if j ≡ 3 (mod 4).
+���
+This computation also shows that, in the limit, the running average of these two
+values will both be 1/2. The subsequence of m, n such that they are separated by
+a fixed integer is density 0 in the sequence of m, n. We can also see that the limit
+of these subsequences, Il(m, m + j), Ir(m, m + j) goes to 1/2 when we take the
+separation integer k to infinity. This ensures via a straightforward limit argument
+that the running average of each piece also goes to 1/2.
+The reason for this can clearly be seen in the explicit computations, as m, n values
+that are close together produce disturbances whose magnitude is not changed when
+m and n are pushed to infinity so long as that separation is maintained. However,
+encountering m and n pairs with that separation becomes less and less likely as m
+and n increase. Numerically we have verified all of this with our solver.
+These two plots are not exactly the same as the direct calculation orders points
+differently. Moreover, the accuracy of the boundary integrals, especially because
+we are dividing them, is not enough to perfectly align these graphs.
+This result establishes that equidistribution fails even on simple subsets of simple
+triangles. In this next section we will expand this result to state these proportions
+need not even be bounded.
+
+10
+H. CHRISTIANSON AND D. PEZZI
+(a) Computed Bottom Left Neumann Data
+for the Right Isosceles
+(b) Plot of Bottom Left Neumann Data using
+Derived Formula
+Figure 2. Bottom Bottom Left Neumann Data Plots: Computed
+and Analytical
+4. Proof of Theorem 3
+In this section, we use the result of Marklof-Rudnick [MR12] to prove Theorem
+3. The idea is to compare the integrals of |h∂xu|2 to those of |u|2 in strips in the
+triangle, and then use the results from [Chr17] to compare the integrals of |h∂xu|2
+to boundary integrals of Neumann data.
+Proof. We drop the subscript and subsequence notation and simply write u for our
+density one subsequence.
+On side F1, the normal derivative is ∂ν = −∂x, and F2 the normal derivative
+is ∂ν = −∂y, and on F3, the tangent derivative is ∂τ = γ−1(a∂x − ∂y) and the
+normal derivative is ∂ν = γ−1(∂x + a∂y). Here γ = (1 + a2)
+1
+2 is the normalizing
+constant. That means that on F1, ∂yu = 0, on F2 ∂xu = 0, and on F3, ∂xu = 1
+γ ∂νu,
+∂yu = a
+γ ∂ν as usual.
+Fix 0 < β < a and δ > 0 independent of h, with δ sufficiently small that
+0 < β − δ2 < β + δ < β + δ + δ2 < a. Let χ(x) be a smooth function satisfying
+
+306090Triangle-BottomLeftNeumannData-200nodes
+6'0
+0.8
+0
+Data
+0.7
+ofNeumann[
+0.6
+Fraction
+0.4
+0.3
+0.2
+0.1
+0
+0
+200
+400
+600
+800
+1000
+1200
+EigenvalueNumberDirectCalculationofBottomLeftData
+6'0
+0.8
+1 of Neumann Data
+0.7 
+0.6
+0.4
+000000000000
+0.3
+0.2
+0.1
+0
+0
+200
+400
+600
+800
+1000
+1200
+EigenvalueNumberENERGY DISTRIBUTION
+11
+0
+β − δ2
+β
+β + δ
+β + δ + δ2
+a
+Figure 3.
+The function χ.
+• χ(x) ≡ 0 for 0 ≤ x ≤ β − δ2,
+• χ(x) ≡ 1 for β + δ + δ2 ≤ x ≤ a,
+• χ′(x) ≥ 0,
+• χ′ = 1
+δ + O(δ) for β ≤ x ≤ β + δ.
+See Figure 3 for a sketch of such a function.
+Let X = χ(x)∂x, and run the usual Rellich commutator argument as in [Chr17]:
+�
+([−h2∆ − 1, X]u)¯udV = −2
+�
+(χ′h2∂2
+xu)¯udV + O(h) = 2
+�
+χ′|h∂xu|2dV + O(h).
+Let Ωβ = Ω ∩ {β − δ2 ≤ x ≤ β + δ + δ2} so that supp χ′ ⊂ Ωβ.
+Further let
+˜Ωβ = Ω ∩ {β ≤ x ≤ β + δ} so that χ′ = δ−1 + O(δ) on ˜Ωβ.
+We write
+2
+�
+Ω
+χ′|h∂xu|2dV = 2
+�
+Ωβ
+χ′|h∂xu|2dV
+≤ 2
+�
+Ωβ
+χ′(|h∂xu|2 + |h∂yu|2)dV
+= 2
+�
+Ωβ
+χ′(−h2∆u)¯udV + O(h)
+= 2
+�
+Ωβ
+χ′|u|2dV + O(h)
+≤ 2(δ−1 + O(δ))
+�
+Ωβ
+|u|2dV + O(h).
+We have
+Area(Ωβ) =
+�
+1 − (β−δ2)
+a
++ 1 − (β+δ+δ2)
+a
+2
+�
+(δ + 2δ2) = (1 − β
+a )δ + O(δ2).
+Hence
+Area(Ωβ)
+Area(Ω) = (1 − β
+a)δ
+a/2
++ O(δ2).
+Then the result of Marklof-Rudnick [MR12] implies
+2(δ−1 + O(δ))
+�
+Ωβ
+|u|2dV = 4(1 − β
+a)
+a
++ O(δ) + o(1),
+
+12
+H. CHRISTIANSON AND D. PEZZI
+so that
+(17)
+�
+([−h2∆ − 1, X]u)¯udV ≤ 4(1 − β
+a)
+a
++ O(δ) + o(1).
+On the other hand,
+�
+([−h2∆ − 1, X]u)¯udV =
+�
+∂Ω
+χ(x)(h∂xu)h∂ν ¯udS.
+On F1, χ(0) = 0 and on F2, ∂x is tangential, so ∂xu = 0. On F3, ∂xu = γ−1∂νu, so
+that
+�
+([−h2∆ − 1, X]u)¯udV = γ−1
+�
+F3
+χ(x)|h∂νu|2dS.
+Putting this together,
+(18)
+γ−1
+�
+F3
+χ(x)|h∂νu|2dS ≤ 4
+a(1 − β
+a ) + O(δ) + o(1).
+We will use (18) to estimate the Neumann data on part of F3. Since χ ≡ 1 on
+{β + δ + δ2 ≤ x ≤ a}, we have
+(19)
+γ−1
+�
+F3∩{β+δ+δ2≤x≤a}
+|h∂νu|2dS ≤ γ−1
+�
+F3
+χ(x)|h∂νu|2dS ≤ 4
+a(1−β
+a )+O(δ)+o(1).
+We now use another commutator type argument to compare the mass of h∂xu
+on the whole triangle to the Neumann data on part of the boundary. To that end,
+let X = (1 − x/a)∂x. Then [−h2∆ − 1, X] = 2a−1h2∂2
+x so that
+�
+Ω
+([−h2∆ − 1, X]u)¯udV = 2
+a
+�
+Ω
+(h2∂2
+xu)¯udV = −2
+a
+�
+Ω
+|h∂xu|2dV.
+On the other hand,
+�
+Ω
+([−h2∆ − 1, X]u)¯udV =
+�
+∂Ω
+(1 − x/a)h∂xuh∂ν ¯udS.
+On F1, x = 0 so X = ∂x = −∂ν. On F2, ∂x is tangential, so that Xu = 0 on F2.
+On F3, we have ∂xu = γ−1∂νu as before. That means
+�
+∂Ω
+(1 − x/a)h∂xuh∂ν ¯udS = −
+�
+F1
+|h∂νu|2dS + γ−1
+�
+F3
+(1 − x/a)|h∂νu|2dS.
+From [Chr17], we know
+�
+F1 |h∂νu|2dS = 2
+a, so that
+�
+∂Ω
+(1 − x/a)h∂xuh∂ν ¯udS = −2
+a + γ−1
+�
+F3
+(1 − x/a)|h∂νu|2dS.
+Rearranging, we have
+(20)
+2
+a
+�
+Ω
+|h∂xu|2dV = 2
+a − γ−1
+�
+F3
+(1 − x/a)|h∂νu|2dS.
+To get an upper bound on the left hand side, we need a lower bound on the
+integral
+γ−1
+�
+F3
+(1 − x/a)|h∂νu|2dS,
+
+ENERGY DISTRIBUTION
+13
+which we do by comparing to the part of the boundary isolated by our cutoff
+function χ. χ(x) ≡ 1 for x ≥ β + δ + δ2, and we have an upper bound on the
+boundary data in this range, not a lower bound. We write
+γ−1
+�
+F3
+(1 − x/a)|h∂νu|2dS
+(21)
+= γ−1
+�
+F3∩{x≥β+δ+δ2}
+(1 − x/a)|h∂νu|2dS
++ γ−1
+�
+F3∩{x≤β+δ+δ2}
+(1 − x/a)|h∂νu|2dS
+≥ (1 − (β + δ + δ2)/a)γ−1
+�
+F3∩{x≤β+δ+δ2}
+|h∂νu|2dS.
+We have
+γ−1
+�
+F3∩{x≤β+δ+δ2}
+|h∂νu|2dS
+= γ−1
+�
+F3
+|h∂νu|2dS − γ−1
+�
+F3∩{x≥β+δ+δ2}
+|h∂νu|2
+and now our upper bound (19) in the region x ≥ β + δ + δ2 is useful. Again using
+the main result from [Chr17], we have
+γ−1
+�
+F3
+|h∂νu|2dS = 2
+a,
+so
+γ−1
+�
+F3∩{x≤β+δ+δ2}
+|h∂νu|2dS
+= γ−1
+�
+F3
+|h∂νu|2dS − γ−1
+�
+F3∩{x≥β+δ+δ2}
+|h∂νu|2
+≥ 2
+a − 4
+a(1 − β
+a ) + O(δ) + o(1).
+Plugging into (21), we have
+γ−1
+�
+F3
+(1 − x/a)|h∂νu|2dS
+≥ (1 − (β + δ + δ2)/a)γ−1
+�
+F3∩{x≤β+δ+δ2}
+|h∂νu|2dS
+≥ (1 − (β + δ + δ2)/a)
+�2
+a − 4
+a(1 − β
+a ) + O(δ) + o(1)
+�
+.
+Combining with (20), we have
+2
+a
+�
+Ω
+|h∂xu|2dV = 2
+a − γ−1
+�
+F3
+(1 − x/a)|h∂νu|2dS
+≤ 2
+a − (1 − (β + δ + δ2)/a)(2
+a − 4
+a(1 − β
+a ) + O(δ) + o(1))
+
+14
+H. CHRISTIANSON AND D. PEZZI
+and rearranging,
+�
+Ω
+|h∂xu|2dV ≤ 1 − (1 − (β + δ + δ2)/a)(1 − 2(1 − β
+a ) + O(δ) + o(1)
+= 1 − (1 − β/a)(1 − 2(1 − β/a)) + O(δ) − o(1).
+(22)
+Optimizing in the variable (1 − β/a) gives (1 − β/a) = 1/4, or
+�
+Ω
+|h∂xu|2dV ≤ 1 − (1/4)(1/2) = 7/8 + O(δ) + o(1)
+as asserted in the Theorem.
+□
+Remark 11. The biggest loss in the proof is from brutally estimating the integral of
+|h∂xu|2 in strips by the integral of |u|2, which is clearly a very crude estimate. It is
+nevertheless interesting to note that if we knew that the integral of |h∂xu|2 in strips
+was half that of |u|2, which would be predicted by quantum ergodicity, the proof still
+does not give the expected estimate on the whole triangle. Indeed, in (17), quantum
+ergodicity would have given 2 (1− β
+a )
+a
++ O(δ) + o(1) instead of 4 (1− β
+a )
+a
++ O(δ) + o(1).
+As in the end of the proof, this would give
+�
+Ω
+|h∂xu|2dV ≤ 1 − (1 − β/a)(1 − (1 − β/a)) + O(δ) − o(1)
+in place of (22). Optimizing again in the variable (1−β/a) yields (1−β/a) = 1/2,
+for a bound of 3/4+O(δ)+o(1). So even if we knew more aboud energy distribution
+compared to distribution, the techniques of proof in this paper give an unsatisfactory
+answer.
+Remark 12. Note this is particular to triangles. Indeed, if Ω = [0, π]2, a basis
+of eigenfunctions consists of umn = cmn sin(mx) sin(ny), where cmn = 2/π is the
+appropriate normalization constant. Let U ⊂ Ω be an open set. We have
+�
+U
+|u|2dV =
+�
+U
+|cmn|2(1/2 − 1/2 cos(2mx))(1/2 − 1/2 cos(2ny))dV
+= π−2
+�
+U
+(1 − cos(2mx) − cos(2nx) + cos(2mx) cos(2ny))dV
+= Area(U)
+Area(Ω) + O(m−1 + n−1).
+On the other hand,
+�
+Ω
+|h∂xu|2dV =
+�
+Ω
+h2m2(4/π2)| cos(mx) sin(ny)|2dV = h2m2,
+and similarly
+�
+Ω |h∂yu|2dV = h2n2.
+Suppose we are interested in {n ≥ Mm} for large M. Then #{m2 + n2 ≤ R2 :
+n ≥ Mm} ∼ R2/M, so has density ∼ 1/M > 0, but
+�
+Ω |h∂xu|2dV ≤ M −2.
+This shows that these eigenfunctions with n ≥ Mm satisfy the spatial equidistri-
+bution as in Marklof-Rudnick:
+�
+U
+|u|2dV = Area(U)
+Area(Ω) + O(Mh)
+but do not have the frequency lower bound property
+�
+Ω |h∂xu|2dV ≥ 1/8.
+
+ENERGY DISTRIBUTION
+15
+5. An Almost Right Isosceles Triangle
+With the right isosceles case taken care of, it is natural to see what happens
+when the domain is perturbed slightly. We will investigate the ’.99 triangle’, or,
+the triangle with vertices {(0, 0), (0, 1), (.99, 0)}.
+Analytical solutions cannot be
+found, but we can use numerical techniques. Using FreeFEM, an online tool for
+using the finite element method to solve PDEs, we have calculated the first 1250
+eigenfunctions and plotted their relevant data.
+Figure 4. .99 Triangle - 1250 Eigenvalues - Y Volume Integral
+and and Bottom Left Neumann plots
+By inspection, these plots have far more going on than the right isosceles case.
+The y volume integral plot is no longer constant, and in fact has some noticeable
+structure. There are at least two, and possibly a third, branches. These branches
+correspond to subsequences of eigenfunctions whose y volume integrals seem to not
+approach 1
+2. There is also a large band with sizable separation from 1
+2. As the energy
+increases, even the less unusual eigenfunctions that have y volume integrals seem
+to be spreading out from the value of 1
+2. These behaviors are discussed numerically
+in the next section.
+The second plot shows the values of Il(m, n) for the .99 triangle, with the adjust-
+ment of the bounds of integration from (0, 1/2) to (0, .495). Some of the structure
+
+99Triangle-YVolumeIntegral-425nodes
+7
+0.9
+0.8
+Calcuated Y Volume Integral
+0.7
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+0
+0
+200
+400
+600
+800
+1000
+1200
+EigenvalueNumber99Triangle-BottomLeftNeumannData-425nodes
+0
+00
+0.9
+：
+0.8
+Data
+200
+0.7
+O
+0
+0
+0.6
+0.4
+0.3
+0.2
+0.1
+0
+0
+200
+400
+600
+800
+1000
+1200
+EigenvalueNumber16
+H. CHRISTIANSON AND D. PEZZI
+of the plots is carried over from the y volume integral case, but it is less coherent.
+Moreover, there seem to be subsequences whose bottom left side Neumann data
+integrals are approaching 1, which indicates that all of the Neumann data is con-
+gregating on one half of the bottom side. This suggests that even a lower bound
+for Neumann data on subsets of the boundary may not be possible, at least not for
+every sequence of eigenfunctions.
+We have verified that the two branches which are apparent in the y volume
+integral plot are comprised of the same eigenfunctions whose bottom left Neumann
+integrals approach 1. Similar pictures for other triangles mentioned throughout
+this paper are in an appendix.
+6. Statistical Analysis of Eigenfunctions on Almost Isosceles Right
+Triangles
+In this section, we introduce several new metrics for measuring how far a sequence
+of eigenfunctions is from having QE or QER type properties.
+6.1. Introducing Running Averages. Statements about quantum ergodicity al-
+low for exceptional zero density subsequences. For the y volume integral, we think
+of 1/2 as signifying quantum ergodicity but, if the domain was truly ergodic, it is
+more accurate to state that every positive density subsequence needs to have a run-
+ning average that converges to 1/2. Or mathematically, for density 1 subsequence
+uik:
+(23)
+aj = 1
+j
+j
+�
+k=1
+�
+T
+|h∂yuik|2dV → 1
+2.
+We can also say something similar about the proportion of Neumann data on
+a given side.
+Here, if the domain was indeed ergodic, for every subsequence of
+proportions, Il(mj, nj), with positive density we have:
+(24)
+1
+j
+j
+�
+k=1
+Il(mj, nj) → 1
+2
+The same is of course true for any data defined similarly on subsets of the
+boundary.
+6.2. Running Averages of Computed Runs. While numerics are never going
+to be able to answer questions like this definitively, they can more accurately set
+expectations. Here are the average values of different metrics from every run done
+throughout this project.
+The ’y volume integral’ and ’Proportion Bottom Left’
+metrics are the ones used throughout this paper. A larger node count represents an
+increase in accuracy, but we found diminishing returns in increasing node counts in
+our numerics. As such, we considered 200 to be sufficient. Values were computed
+for the first 1250 eigenfunctions.
+
+ENERGY DISTRIBUTION
+17
+Triangle
+Nodes
+y volume integral
+Proportion of Bottom Left
+.99
+425
+.4998
+.5083
+.98
+200
+.4996
+.5098
+.97
+200
+.4996
+.5103
+.96
+200
+.4993
+.5097
+.95
+200
+.4992
+.5100
+The overall trend is consistent across metrics. The further we get from the right
+isosceles triangle, the farther the metrics get from the values quantum ergodicity
+would predict. This is not enough evidence to suggest that these averages converge
+to a value other than what would be expected if the domain was ergodic, but it does
+heavily suggest that convergence is at least slower the farther away from isosceles
+the triangle is.
+6.3. Percentage of Eigenfunctions Approach. The issue with the methods
+previously described in this chapter is that they do not get to the heart of what
+we want. Running averages can be influenced, especially at these frequencies, by
+density zero subsequences which are interesting but not definitive evidence that
+the domain itself is not ergodic. In service of trying to determine whether these
+experiments would cause us to expect a positive density subsequence that converges
+to an unexpected value, we instead shift our focus to percentages of eigenfunctions.
+Statements about the density of sequences are extensions of the familiar discrete
+concept of percentages. They are statements about how common we would expect
+that particular subsequence to be. A density 1 subsequence, in the high-frequency
+limit, would be expected to appear for almost every value. As these are limits,
+there is substantial wiggle room.
+We can use this concept to develop metrics that could indicate whether posi-
+tive density subsequences of the desired properties exist. Suppose we thought the
+running average of the y volume integrals for the .99 triangle converged to a value
+less than .5. Then it would be sufficient to show for every finite N, some fixed
+ϵ > 0, and some other fixed δ > 0, that the percentage of the first N eigenfunctions
+which have an y volume integral less than .5 − ϵ is larger than δ for every N. If
+this condition was met, than the subsequence of all eigenfunctions whose y volume
+integral is less than .5 − ϵ would have a density greater than δ. This would show
+that the domain itself was not ergodic.
+Of course, there is nothing special about viewing the percentage of eigenfunctions
+below a certain threshold. Because we only need a subsequence of positive density,
+we can consider all eigenfunctions that have y volume integrals sufficiently far away
+from .5. In the interest of having a metric that is equally valid regardless of the
+distribution of y volume integral values, we consider the running percentage of
+eigenfunctions such that
+(25)
+���
+�
+T
+|h∂yu|2 − .5
+��� > ϵ
+for varying tolerances ϵ. The values for a selection of runs are in the following table.
+
+18
+H. CHRISTIANSON AND D. PEZZI
+Figure 5. Running Percentage Graph - Shows monotonic and
+asymptotic behavior
+Figure 6. Running Percentage Graph - Shows monotonic and
+asymptotic behavior
+Triangle
+ϵ = .01
+ϵ = .005
+ϵ = .001
+.99
+80.32
+84.64
+98.8
+.98
+82.9
+93.0
+99.3
+.97
+89.1
+95.7
+99.6
+.96
+91.0
+95.6
+99.04
+.95
+92.7
+96.6
+99.52
+Perhaps more interesting than the exact numerical values are the trends. All of
+the graphs for all three thresholds for the .99, .98, .97, .96 and .95 triangles have
+the same fundamental shape: increasing with a vertical asymptote.
+6.4. Establishing Triangles with Different Behavior. An interesting test case
+is the 30-60-90 triangle. Despite having the spatial equidistribution property from
+being a rational planar polygon, it is known to be integrable. This triangle has
+
+pointNineNineTriangle: Percent of dy integrals outside 0.01 threshhold
+100
+90
+80
+70
+09
+50
+40
+30
+20
+10
+0
+0
+200
+400
+600
+800
+1000
+1200
+1400pointNineNineTriangle: Percent of dy integrals outside 0.0o5 threshhold
+100
+90
+80
+70 
+09
+50 
+40
+30
+20
+10
+0
+0
+200
+400
+600
+800
+1000
+1200
+1400ENERGY DISTRIBUTION
+19
+Figure 7. Running Percentage Graph - Shows monotonic and
+asymptotic behavior
+a lot of symmetries, reflecting it over the y-axis gives the equilateral triangle for
+example, which is what leads to its integrability. By looking at triangles that are
+close to the 30-60-90, we can see how sensitive these numerics are.
+We ran two runs with a bottom length of .575 and .58. The bottom length of
+the 30-60-90 is
+1
+√
+3 ≈ .5774, so these other triangles are close the the 30-60-90 but
+do not enjoy the geometric symmetries that have such a profound effect on the
+eigenfunctions. They produced the following results:
+Triangle
+ϵ = .01
+ϵ = .005
+ϵ = .001
+.58
+42.6
+65.0
+96.8
+30-60-90
+11.7
+16.4
+30.9
+.575
+41.8
+61.1
+97.2
+Not only are the percentages noticeably lower than the other triangles, the shape
+of the running percentage scatter plot indicates that these numbers are decreasing
+significantly as the number of eigenvalues increases. This is the type of behavior
+that would be expected for an ergodic domain, but we see behaviors more in line
+with the previously discussed runs for the two triangles that are close to the 30-60-
+90. This complicates our interpretation, as we have a non-ergodic triangle that is
+displaying behavior that would be expected of an ergodic domain.
+7. Accuracy and Sanity Checks
+
+pointNineNineTriangle: Percent of dy integrals outside 0.001 threshhold
+100
+90
+80
+70
+09
+50
+40
+30
+20
+10 
+0
+J
+0
+200
+400
+600
+800
+1000
+1200
+140020
+H. CHRISTIANSON AND D. PEZZI
+(a) 30-60-90 - 450 Nodes - 1000 Eigenvalues
+(b) .58 triangle - 200 Nodes - 1250 Eigenval-
+ues
+7.1. Mesh Convergence Test. Confidence in our numerics increases if we can
+show convergence in accuracy metrics as our mesh is refined. To test this, we chose
+two metrics: one for the eigenvalue and one for the eigenfunctions.
+The maximum eigenvalue difference is simply the largest difference between
+eigenvalues computed on the different meshes. The L2 running average is the run-
+ning average of the L2 norm of the difference between the eigenfunctions computed
+on different meshes. To evaluate this difference, the higher accuracy function is
+interpolated on the coarser mesh. This adds another source of inaccuracy.
+We compared the 256 node calculations to the 128, 64, and 32 node calculations
+for the .99 triangle. The first 1000 eigenvalues and eigenfunctions were computed.
+The table below shows clear convergence on both metrics.
+Comparison
+Max Eval Diff.
+L2 Running Avg.
+256 and 128
+8.87
+.0091
+256 and 64
+122.23
+.0838
+256 and 32
+377.87
+.4762
+The 1000th Eigenvalue has a magnitude of around 30,000, so a maximum differ-
+ence of 8.87 corresponds to about a .03% difference. This shows we are not gaining
+a substantial amount of accuracy doubling the perimeter node count once we pass
+
+306090Triangle:Percentofdyintegralsoutside0.01threshhold
+1 [
+0.9
+0.8
+0.7
+0.6
+0.5
+0.4
+0.3
+0.2 
+0.1
+0
+100
+200
+300
+400
+500
+600
+700
+800
+006
+1000pointFiveEightTriangle: Percent of dy integrals outside 0.01 threshhold
+0.9
+0.8
+0.7
+0.6
+0.5
+0.4
+0.3
+0.2 
+0
+200
+400
+600
+800
+1000
+1200
+140021
+a certain threshold. This gives us confidence that our numerical experiment is well
+behaved.
+7.2. Reported Errors. FreeFEM itself can also report errors. It does this in 3
+types, the relative error, absolute error, the backward error. All of these errors are
+generally monotonically increasing, so we will just report the error on the 1250th
+eigenvalue.
+Error Type
+Value
+Absolute Error
+8.77e-8
+Relative Error
+3.04e-15
+Backwards Error
+7.292e-12
+Appendices
+A. Volume and Boundary Data for Near Isosceles Triangles
+Figure 9. .99 Triangle - 1250 Eigenvalues - Y Volume Integral
+and Bottom Left Neumann plots
+
+99Triangle-YVolumeIntegral-425nodes
+7
+0.9
+0.8
+Calcuated Y Volume Integral
+0.7
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+0
+0
+200
+400
+600
+800
+1000
+1200
+EigenvalueNumber99Triangle-BottomLeftNeumannData-425nodes
+0
+00
+0.9
+：
+0.8
+Data
+200
+0.7
+O
+0
+0
+0.6
+0.4
+0.3
+0.2
+0.1
+0
+0
+200
+400
+600
+800
+1000
+1200
+EigenvalueNumber22
+Figure 10. .98 Triangle - 1250 Eigenvalues - Y Volume Integral
+and Bottom Left Neumann plots
+References
+[CdV85] Y. Colin de Verdi`ere. Ergodicit´e et fonctions propres du laplacien. Comm. Math. Phys.,
+102(3):497–502, 1985.
+[Chr17] Hans Christianson. Equidistribution of Neumann data mass on triangles. Proc. Amer.
+Math. Soc., 145(12):5247–5255, 2017.
+[Chr19] Hans Christianson. Equidistribution of Neumann data mass on simplices and a simple
+inverse problem. Math. Res. Lett., 26(2):421–445, 2019.
+[CTZ12] Hans Christianson, John Toth, and Steve Zelditch. Quantum ergodic restriction for
+cauchy data: Interior QUE and restricted QUE. preprint, 2012.
+[Has10] Andrew Hassell. Ergodic billiards that are not quantum unique ergodic. Ann. of Math.
+(2), 171(1):605–618, 2010. With an appendix by the author and Luc Hillairet.
+[HZ04]
+Andrew Hassell and Steve Zelditch. Quantum ergodicity of boundary values of eigenfunc-
+tions. Comm. Math. Phys., 248(1):119–168, 2004.
+[Lin06] Elon Lindenstrauss. Invariant measures and arithmetic quantum unique ergodicity. Ann.
+of Math. (2), 163(1):165–219, 2006.
+[MR12] Jens Marklof and Ze´ev Rudnick. Almost all eigenfunctions of a rational polygon are
+uniformly distributed. J. Spectr. Theory, 2(1):107–113, 2012.
+[Shn74] A. I. Shnirelman. Ergodic properties of eigenfunctions. Uspehi Mat. Nauk, 29(6(180)):181–
+182, 1974.
+[TZ12]
+J.A. Toth and S. Zelditch. Quantum ergodic restriction theorems, i: interior hypersurfaces
+in domains with ergodic billiards. Annales Henri Poincar´e, 13:599–670, 2012.
+
+98Triangle-YVolumeIntegral-200 nodes
+0.9
+0.8
+0.7 
+0.6
+00000000
+QQ0000000
+0
+0.5
+0.4
+0.3
+0.2
+0.1
+0
+0
+200
+400
+600
+800
+1000
+1200
+EigenvalueNumber98Triangle-BottomLeftNeumannData-200nodes
+00
+0.9
+0000
+X
+50.000000008
+oooooooooo
+00000
+0.8
+Data
+0.7
+0
+Q
+0
+0
+0.6
+0.5
+0.4
+ee
+0.3
+0.2
+0.1
+0
+0
+200
+400
+600
+800
+1000
+1200
+EigenvalueNumber23
+Figure 11. .97 Triangle - 1250 Eigenvalues - Y Volume Integral
+and Bottom Left Neumann plots
+[TZ13]
+John A. Toth and Steve Zelditch. Quantum ergodic restriction theorems: manifolds with-
+out boundary. Geom. Funct. Anal., 23(2):715–775, 2013.
+[Zel87]
+Steven Zelditch. Uniform distribution of eigenfunctions on compact hyperbolic surfaces.
+Duke Math. J., 55(4):919–941, 1987.
+[ZZ96]
+Steven Zelditch and Maciej Zworski. Ergodicity of eigenfunctions for ergodic billiards.
+Comm. Math. Phys., 175(3):673–682, 1996.
+(H. Christianson) Department of Mathematics, University of North Carolina.
+Email address: [email protected]
+(D. Pezzi) Department of Mathematics, Johns Hopkins.
+Email address: [email protected]
+
+97Triangle-YVolume Integral-200 nodes
+7
+0.9
+0000.
+00
+000
+0.8
+0 00000
+0.7
+0
+0
+0.6
+8
+0.4
+0.3
+0.2
+0.1
+0
+0
+200
+400
+600
+800
+1000
+1200
+EigenvalueNumber.97Triangle-BottomLeftNeumannData-200nodes
+Q
+000&0
+0.9
+0
+0.8
+000
+6
+0
+Data
+0
+00
+门
+0
+0.7
+0
+00
+0
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+0
+0
+200
+400
+600
+800
+1000
+1200
+EigenvalueNumber24
+Figure 12. .96 Triangle - 1250 Eigenvalues - Y Volume Integral
+and Bottom Left Neumann plots
+
+96Triangle-YVolumeIntegral-200 nodes
+0.9
+0.8
+Integral
+000000
+0.7
+0
+000
+YVolume
+b
+0.6
+0.5
+Calcuated
+0.4
+0.3
+0.2
+0.1
+0
+0
+200
+400
+600
+800
+1000
+1200
+EigenvalueNumber96Triangle-BottomLeftNeumannData-200nodes
+0000000
+0
+0
+0.9
+0
+0
+0.8
+0
+00
+0
+Data
+0
+0
+00
+00
+00
+00080
+0.7
+00
+000
+0
+00
+0
+8
+0.6
+000
+00
+0.5
+0.4
+0.3
+)
+0.2
+0.1
+0
+0
+200
+400
+600
+800
+1000
+1200
+EigenvalueNumber25
+Figure 13. .95 Triangle - 1250 Eigenvalues - Y Volume Integral
+and Bottom Left Neumann plots
+
+95Triangle-YVolumeIntegral-200nodes
+0.9
+0.8
+Integral
+000000
+0.7
+0
+000
+YVolume
+b
+0.6
+0.5
+Calcuated
+0.4
+0.3
+0.2
+0.1
+0
+0
+200
+400
+600
+800
+1000
+1200
+EigenvalueNumber.95Triangle-BottomLeftNeumannData-200nodes
+C
+QQ
+0
+0
+0.9
+0Q
+0.8
+e
+Data
+0
+0.7
+0)
+00
+0.6
+0.5
+0.4
+0.3 
+0.2
+0.1
+0
+0
+200
+400
+600
+800
+1000
+1200
+EigenvalueNumber

HdAyT4oBgHgl3EQf5frD/content/tmp_files/2301.00807v1.pdf.txt

@@ -0,0 +1,1552 @@
+1 
+ 
+New Analytical Approach Based on Transfer Matrix Method (TMM) for 
+Study of Tunable Plasmonic Modes in Graphene-Based Heterostructures    
+ 
+ 
+Mohammad Bagher Heydari 1,*, Majid Karimipour 2, Morteza Mohammadi Shirkolaei 3 
+ 
+1,* School of Electrical Engineering, Iran University of Science and Technology (IUST), Tehran, Iran 
+2 Department of Electrical Engineering, Arak University of Technology, Arak, Iran 
+3 Department of Electrical Engineering, Shahid Sattari Aeronautical University of Science and Technology, Tehran, 
+Iran 
+  
+*Corresponding author: [email protected] 
+ 
+ 
+Abstract: This paper aims to study the reflection characteristics of optical beams in a hybrid graphene-hexagonal 
+Boron Nitride (hBN)-graphene structure, which has been located on SiO2-Si layers. An analytical model is presented 
+to derive the reflection characteristics by using the transfer matrix method. The upper Reststrahlen band has been 
+chosen as the studied frequency range. It is shown that the characteristics of the reflected beam can be effectively 
+controlled by varying the chemical potential of graphene sheets. The obtained results represent a high value of the 
+reflected group delay (𝜏𝑟 = 15.3 𝑝𝑠) at the frequency of 24.9 THz. The presented investigation will be helpful to 
+control the group delays of reflected beams and can be utilized for the design of innovative graphene-hBN devices in 
+the mid-infrared wavelengths.         
+ 
+Key-words: Graphene, hBN, plasmon, phonon, analytical   
+  
+ 
+1. Introduction 
+Nowadays, graphene has attracted immense interest among scientists in nano-electronics and THz applications [1]. It 
+has exceptional features in the mid-infrared region, such as high thermal and optical conductivities, which can be 
+explored in many research areas of physics and chemistry. The optical conductivity of graphene opens the way to 
+flexibly control and adjust the propagation features of plasmonic components such as couplers [2-4], filters [5-7], 
+resonators [8-10], circulators [11-14], waveguides [15-24], sensing [25-30], and imaging [31, 32]. Graphene-based 
+waveguides have various structures such as planar [16, 24, 33-63], cylindrical [44, 64-69], and elliptical structures 
+[23, 70-72]. Combining graphene with other Van der Waals materials can be interesting because hybrid 
+heterostructures have compound properties of both materials. Hexagonal Boron Nitride (hBN) is one of the famous 
+van der Waals materials in the mid-infrared frequencies in which its permittivity function shows two kinds of phonon 
+modes [73-75]. The hybridization of graphene with this material can generate and support new types of propagating 
+modes called “coupled phonon-plasmon polaritons” [76-82].                      
+Controlling the group delay of the optical beam is one of the debated topics in recent years and it has many 
+fascinating applications such as optical buffers and delay lines [83]. In optics, some methods have been introduced 
+and reported to achieve high levels of the group delays such as the spin Hall effect [84] and photonic crystal [85]. 
+However, in the mid-infrared region, there is limited research related to obtaining large, tunable group delay [86]. One 
+of the interesting ways to flexibly obtain and vary the group delay in this band is the usage of hybrid heterostructures.  
+Here, we propose a hybrid heterostructure composed of graphene-hBN-graphene layers. Two graphene sheets 
+have been utilized in our system to increase the degree of freedom to adjust the reflection characteristics more flexibly. 
+The whole structure is illuminated by a TM-polarized beam with an incident angle of θ. The analytical expressions 
+
+2 
+ 
+are derived for the calculation of the reflection characteristics of the structure by using the transfer matrix method. A 
+large value of reflected group delay, i.e. 𝜏𝑟 = 15.3 𝑝𝑠, is achievable at the frequency of 24.9 THz.   
+It is worthwhile to compare the proposed structure over similar configurations reported in the literature [87-92] 
+to give a better insight into the flexibility and superiority of the presented heterostructure. In [87], the authors have 
+studied phonon plasmon polariton modes in two Reststrahlen bands for multilayered graphene-hBN metamaterials 
+and have reported the dispersion diagrams for these modes. A similar study is done in [88], where hyperbolic plasmon-
+phonon modes are examined by nano-infrared imaging. No value for reflected group delays is reported in [87, 88] 
+because their focus is on the existence of these modes and the investigation of the propagating features. In [89], an 
+electromagnetic absorber based on the graphene-hBN hyper crystal is proposed and authors have obtained a perfect 
+absorber near 750 cm-1 at the incident angle of θ=670. A new mechanism for the directed excitation of plasmon-phonon 
+modes is suggested in [90] and negative refraction of hybrid phonon modes is presented by the same research group 
+in [91]. In [92], the reflected group delay is reported as τ=13.97 ps for the chemical potential of 0.25 eV for their 
+structure while our obtained result is about 15.3 ps at the frequency of 24.9 THz. Therefore, our proposed structure is 
+tunable and can change effectively the characteristics of the reflected beam by varying the chemical potential of 
+graphene sheets.  
+The remainder of the paper is organized as follows. In section 2, after introducing the proposed structure, 
+mathematical expressions will be presented for the reflected group delay. Then, in section 3, the analytical results are 
+reported and investigated more precisely. It will be shown that the hybrid structure can flexibly control the reflected 
+beam by changing the chemical potential of graphene sheets. Finally, section 4 concludes the article.                
+ 
+  
+2. The Proposed Heterostructure and its Analytical Model                
+Fig. 1 shows the configuration of the studied heterostructure, where the hBN layer is sandwiched between two 
+graphene sheets and the composite structure is located on SiO2-Si layers. The whole structure is illuminated by a TM-
+polarized beam with an incident angle of θ. There is no layer below the Si layer. The conductivity of each graphene 
+sheet can be modeled by the following relation [93]:  
+
+
+
+
+,1,2
+2
+2
+,1,2
+,1,2
+1,2
+2
+,1,2
+2
+(
+j2 )
+,
+, ,
+2
+1
+4
+2
+(
+j2 )
+(
+j2 )
+B
+c
+c
+c
+B
+c
+B
+c
+K T
+je K T
+je
+T
+Ln
+Ln
+e
+K T
+
+
+
+
+
+ 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+                         (1)                                 
+In (1),  𝛤  is the scattering rate, 𝑇 is the temperature, and 𝜇𝑐,1,2 is the chemical potential of each graphene. Furthermore, 
+ℎ is the reduced Planck’s constant, 𝐾𝐵 is Boltzmann’s constant, ω is radian frequency, and 𝑒 is the electron charge in 
+this relation.     
+ 
+Figure. 1. The schematic of the studied structure.              
+Graphene 1  
+𝑑  
+𝜎1  
+SiO2  
+ɛ𝑆𝑖𝑂2 
+Air 
+hBN  
+𝑡  
+𝑧  
+𝑥  
+𝑦  
+ɛ ℎ𝐵𝑁 
+ɛ𝑆𝑖 
+Si 
+Graphene 2  
+𝜎2  
+𝜃 
+
+3 
+ 
+ 
+hBN is a polar dielectric, supporting two phonon modes related to hyperbolicity, with the following permittivity 
+tensor [74]:  
+ 
+
+
+
+
+
+
+2
+2
+,
+,
+,
+,
+2
+2
+,
+.
+LO m
+TO m
+m
+m
+m
+TO m
+m
+j
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+                                                                                                        (2) 
+In (2), 𝑚 = ‖ 𝑜𝑟 ⊥ is related to the transverse and z-axis, respectively. Moreover, 𝜔𝐿𝑂, 𝜔𝑇𝑂 show the LO and TO 
+phonon frequencies, respectively, in which each frequency has two values in the upper and lower Reststrahlen 
+bands: 𝜔𝐿𝑂,⊥ = 24.9 𝑇𝐻𝑧, 𝜔𝑇𝑂,⊥ = 23.4 𝑇𝐻𝑧, 𝜔𝐿𝑂,‖ = 48.3 𝑇𝐻𝑧, 𝜔𝑇𝑂,‖ = 41.1 𝑇𝐻𝑧. In (2), 𝛤𝑚 is a damping factor 
+(𝛤⊥ = 0.15 𝑇𝐻𝑧, 𝛤‖ = 0.12 𝑇𝐻𝑧) and ɛ𝑚 is related to the high-frequency permittivity (ɛ∞,⊥ = 4.87, ɛ∞,‖ = 2.95) [74]. 
+In fig. 2, the dielectric function of hBN permittivity is depicted, which shows the lower and upper Reststrahlen bands. 
+ 
+Figure. 2. The permittivity of hBN versus frequency. The lower and upper Reststrahlen bands are shown in this 
+figure.    
+ 
+To calculate the reflection coefficient and the reflected group delay, the transfer matrix method can be utilized. 
+For various regions, TM-polarized waves (p-polarized waves) can be written as follows: 
+
+
+
+
+
+
+
+
+,1
+,1
+,2
+,2
+,3
+,3
+,4
+,4
+,1
+1
+1
+,2
+2
+2
+,3
+3
+3
+,4
+4
+4
+0
+0
+z
+z
+x
+z
+z
+x
+z
+z
+x
+z
+z
+x
+ik
+z
+ik
+z
+ik x
+y
+ik
+z
+ik
+z
+ik x
+y
+ik
+z
+ik
+z
+ik x
+y
+ik
+z
+ik
+z
+ik x
+y
+H
+a e
+b e
+e
+z
+H
+a e
+b e
+e
+z
+t
+H
+a e
+b e
+e
+t
+z
+t
+d
+H
+a e
+b e
+e
+t
+d
+z
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ 
+
+
+
+
+                                                                                           (3) 
+Thus, the transfer matrix of the whole structure is obtained:   
+1
+4
+1
+4
+.
+a
+a
+M
+b
+b
+
+ 
+
+
+ 
+
+
+ 
+
+
+                                                                                                                                                               (4) 
+Where 
+1
+2
+2
+2
+3
+3
+3
+4
+. .
+.
+.
+M
+D
+P D
+P D
+
+
+
+
+                                                                                                                                   (5) 
+From Air to hBN, the transmission matrix is written as follows:     
+1
+2
+1
+1
+1
+1
+1
+2
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+D
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+                                                                                                               (6) 
+ 
+In (6), the following parameters have been defined: 
+
+4 
+ 
+0
+2
+1
+z
+TM
+z
+k
+k
+
+
+
+
+                                                                                                                                                                    (7) 
+1
+2
+0
+z
+TM
+k
+
+
+  
+
+
+                                                                                                                                                               (8) 
+Moreover, the wave number component of the incident beam in the z-direction can be obtained by (it is supposed that 
+the direction of the incident angle is θ): 
+1
+0 cos
+z
+k
+k
+
+
+                                                                                                                                                              (9) 
+
+
+2
+2
+2
+0
+sin
+z
+k
+k
+
+
+
+
+
+
+
+
+                                                                                                                                         (10) 
+Now, the propagation matrix of the plasmonic wave inside the hBN layer is obtained by the following matrix (the 
+thickness of the hBN medium is assumed to be t):  
+2
+2
+2
+0
+0
+z
+z
+jk
+t
+jk
+t
+e
+P
+e
+
+
+
+
+
+ 
+
+
+
+                                                                                                                                                      (11) 
+When the beam inside the hBN medium reaches the surface of the second graphene sheet, the transmission matrix 
+from the hBN to the SiO2 layer is expressed as: 
+2
+3
+1
+1
+1
+1
+1
+2
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+D
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+                                                                                                             (12) 
+Where the following parameters have been utilized in (12): 
+2
+3
+2
+z
+TM
+SiO
+z
+k
+k
+
+
+
+
+
+
+                                                                                                                                                              (13) 
+2
+2
+3
+0
+z
+TM
+SiO
+k
+
+
+ 
+
+
+
+                                                                                                                                                                    (14) 
+Similar to the propagation matrix of the beam inside the hBN medium, the propagation matrix inside the SiO2 layer 
+can be written as (the thickness of the SiO2 layer is assumed to be d): 
+3
+3
+3
+0
+0
+z
+z
+jk d
+jk d
+e
+P
+e
+
+
+
+
+
+ 
+
+
+
+                                                                                                                                                  (15) 
+Finally, as the beam reaches the border of SiO2-Si layers, the transmission matrix can be expressed as: 
+3
+4
+1
+1
+1
+1
+1
+2
+TM
+TM
+TM
+TM
+D
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+                                                                                                                                         (16) 
+In (16), the following parameters have been defined: 
+2
+4
+3
+SiO
+z
+TM
+Si
+z
+k
+k
+
+
+
+ 
+                                                                                                                                                                 (17) 
+By calculating the elements of the transfer matrix, the reflection coefficient and the reflectance are derived by:               
+ 
+ 
+
+
+21
+11
+exp
+r
+M
+r
+r
+j
+M
+
+
+
+
+
+                                                                                                                                 (18) 
+ 
+2
+R
+r 
+
+                                                                                                                                                                           (19) 
+For the proposed structure, 𝑀21 and 𝑀11 are calculated:  
+
+5 
+ 
+
+ 
+ 
+
+
+ 
+ 
+
+
+ 
+ 
+
+
+ 
+ 
+
+11
+3
+2
+3
+2
+3
+2
+3
+2
+1
+. 1
+. 1
+.
+.
+1
+. 1
+. 1
+.
+.
+1
+. 1
+. 1
+.
+.
+1
+. 1
+. 1
+.
+.
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+z
+z
+z
+z
+z
+z
+z
+z
+jk d
+jk
+t
+jk d
+jk
+t
+jk d
+jk
+t
+jk d
+jk
+t
+M
+e
+e
+e
+e
+e
+e
+e
+e
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+                                                                 (20) 
+
+ 
+ 
+
+
+ 
+ 
+
+
+ 
+ 
+
+
+ 
+ 
+
+21
+3
+2
+3
+2
+3
+2
+3
+2
+1
+. 1
+. 1
+.
+.
+1
+. 1
+. 1
+.
+.
+1
+. 1
+. 1
+.
+.
+1
+. 1
+. 1
+.
+.
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+TM
+z
+z
+z
+z
+z
+z
+z
+z
+jk d
+jk
+t
+jk d
+jk
+t
+jk d
+jk
+t
+jk d
+jk
+t
+M
+e
+e
+e
+e
+e
+e
+e
+e
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+                                                                (21) 
+Here, if we suppose that the incident pulse is a Gaussian beam with the central and half-width of 𝜔0, 𝜏0, respectively: 
+
+
+
+
+
+
+2
+2
+0
+0
+0
+0,
+exp
+2
+exp
+iE
+t
+A
+t
+i
+t
+
+
+
+
+
+                                                                                                              (22) 
+It should be noted that the corresponding Fourier spectrum of (22) is: 
+
+
+
+
+
+
+2
+2
+2
+0
+0
+0
+0
+0,
+exp
+2
+2
+i
+A
+E
+
+
+
+
+
+
+
+
+
+                                                                                                                (23) 
+and thus the group delay is not a function of time (t) because all relations are written in the Fourier space (ω). Then, 
+the reflected group delay is obtained:                             
+ 
+r
+r
+c
+ 
+
+
+
+
+
+
+
+
+ 
+
+
+
+
+                                                                                                                                                            (24) 
+In (24), 𝜔𝑐 is the carrier frequency. Now, our model is completed for the proposed heterostructure. In what follows, 
+we will investigate the analytical results of the above mathematical relations.          
+ 
+ 
+3.  Results and Discussions  
+This section reports the analytical results of the proposed structure. In these results, the chemical potential of graphene 
+sheets is supposed to be 𝜇𝑐,1 = 0.2 𝑒𝑣, 𝜇𝑐,2 = 0.3 𝑒𝑣, respectively. The temperature is 𝑇 = 300 𝐾 and the relaxation 
+time is assumed to be 𝜏 = 0.45 𝑝𝑠. Both graphene layers have the similar thickness 𝛥1 = 𝛥2 = 𝛥 = 0.33 𝑛𝑚. The 
+parameters of the hBN medium have been given in the previous section. Moreover, the geometrical parameters are 𝑡 =
+100𝑛𝑚, 𝑑 = 150𝑛𝑚. The permittivity constant of SiO2 and Si layers are assumed to be 2.09 and 11.9, respectively. 
+The incident angle is 𝜃 = 450.        
+As explained before, there are two phonon modes in the hBN medium (see fig. 2) that are related to hyperbolicity: 
+out-of-plane and in-plane phonon modes, which lead to two various Reststrahlen bands. First, let us consider the 
+reflected group delays in three different frequency ranges: 24.75-25 THz (in the lower Reststrahlen band), 36.15-37.5 
+THz (in the middle band, i.e. between the lower and upper Reststrahlen bands) and 41.1-41.6 THz (in the upper 
+Reststrahlen band). It should be emphasized that the resonance frequency at 24.855 THz in fig. 3 (a) is not related to 
+the lower TO phonon frequency (𝜔𝑇𝑂,⊥ = 23.4 𝑇𝐻𝑧). As observed in Fig. 3 (a), the group delay shows the metal-like 
+behavior in the lower Reststrahlen band which can be enhanced. While the reflected group delay in other frequency 
+ranges (fig. 3 (b), (c)) cannot be enhanced because it has negligible values which originate from the high values of 
+hBN losses (imaginary part of hBN dielectric function). Therefore, we only focus on the first frequency window, i.e. 
+24.15-25 THz (the lower Reststrahlen band), in the following results. It should be noted that the reflected group delay 
+in our structure originates from the Lorentz resonance mechanism.     
+
+6 
+ 
+ 
+ 
+ 
+Figure. 3. Reflected group delay versus frequency at three frequency windows: (a) 24.75 THz-25 THz (in the lower 
+Reststrahlen band), (b) 36.15 THz-37.5 THz (in the middle band, i.e. between the lower and upper Reststrahlen 
+bands), (c) 41.1 THz-41.6 THz (in the upper Reststrahlen band). The chemical potential of graphene layers is 
+supposed to be 𝜇𝑐,1 = 0.2 𝑒𝑣,  𝜇𝑐,2 = 0.3 𝑒𝑣. The thickness of the hBN and SiO2 layers are 100 nm and 150 nm, 
+respectively. The incident angle is 𝜃 = 450.  
+ 
+In the previous section, we analytically obtained the elements of the transfer matrix for the proposed 
+heterostructure. The studied structure is a tunable device in which its reflection characteristics can be varied by 
+changing the chemical potential. Fig. 4 represents the variations of reflectance, the group delay, and the reflected phase 
+as a function of frequency for various values of chemical potential. As noted before, the frequency range is 24.75-25 
+THz. The incident angle is chosen 𝜃 = 450. It can be found from fig. 4 (a) that the reflectance has a dip of around 
+24.88 THz and its place can be varied as the values of chemical potential change. Around 24.88 THz, the sign of 
+reflected phase changes, as seen in fig. 4 (b). Meanwhile, the peak of the reflected group delay varies for various 
+values of chemical potential. A large value of reflected group delay, i.e. 𝜏𝑟 = 12.2 𝑝𝑠, is reported for the chemical 
+potentials of  𝜇𝑐,1 = 0.2 𝑒𝑣,  𝜇𝑐,2 = 0.8 𝑒𝑣 at the frequency of 24.85 THz.     
+Fig. 5 shows the reflected group delay of the optical beam as a function of frequency for various values of 
+graphene thickness. In this diagram, it is supposed that both graphene sheets have similar thicknesses (𝛥1 = 𝛥2 = 𝛥). 
+The chemical potential of graphene layers are remained fixed: 𝜇𝑐,1 = 0.2 𝑒𝑣,  𝜇𝑐,2 = 0.3 𝑒𝑣. As the number of 
+graphene layers increases (the thickness increases), the peak value of the group delay increases, as observed in fig. 5. 
+Furthermore, one can see that the maximum peak shifts to the higher frequencies as the thickness increases. For thicker 
+graphene sheets, a high value of group delay is achievable. For instance, the reflected group delay of 15.3ps is obtained 
+for the thickness of 𝛥1 = 𝛥2 = 𝛥 = 1 𝑛𝑚 at the frequency of 24.9 THz.   
+It is worth to be mentioned that the thickness of various layers in the proposed heterostructure can change the 
+reflection characteristics of the reflected beam. One of these parameters is the thickness of the hBN layer, where its 
+variations have been depicted in fig. 6. The chemical potential of graphene layers are 𝜇𝑐,1 = 0.2 𝑒𝑣,  𝜇𝑐,2 = 0.3 𝑒𝑣. 
+Both graphene layers have the similar thickness 𝛥1 = 𝛥2 = 𝛥 = 0.33 𝑛𝑚. Other parameters have remained fixed. It 
+
+7 
+ 
+can be seen from fig. 5 that the maximum point of the reflected group delay shifts to higher frequencies as the thickness 
+of the hBN medium increases. However, the changes are slight because the thickness of the hBN layer varies from 
+100nm to 120 nm.   
+ 
+ 
+ 
+ 
+Figure.4. The variations of reflectance, the reflected phase, and the reflected group delay as a function of frequency 
+in the lower Reststrahlen band, for various values of the chemical potential of graphene layers. The thickness of the 
+hBN and SiO2 layers are 100 nm and 150 nm, respectively. The incident angle is 𝜃 = 450.   
+ 
+Figure. 5. The reflected group delay versus frequency for different values of graphene thickness. It is supposed that 
+both graphene layers have similar thicknesses (𝛥1 = 𝛥2 = 𝛥).  The chemical potential of graphene layers is 
+supposed to be 𝜇𝑐,1 = 0.2 𝑒𝑣,  𝜇𝑐,2 = 0.3 𝑒𝑣. The thickness of the hBN and SiO2 layers are 100 nm and 150 nm, 
+respectively. The incident angle is 𝜃 = 450.   
+ 
+As a final point, we investigate the influence of SiO2 thickness on the reflected group delay. As derived in relations 
+(16)-(21), the thickness of the SiO2 layer can change the characteristics of the reflected beam. One can observe from 
+fig. 7 that as the SiO2 thickness varies from 150nm to 200 nm, the maximum peak changes, and its frequency shifts to 
+higher frequencies. The presented study on the reflection characteristics of the reflected beam of the proposed 
+heterostructure is useful for potential applications such as the design of optical delay lines and optical buffers.   
+
+8 
+ 
+ 
+Figure. 6. The reflected group delay versus frequency for different values of hBN thickness. It is supposed that both 
+graphene layers have similar thicknesses (𝛥1 = 𝛥2 = 𝛥 = 0.33 𝑛𝑚).  The chemical potential of graphene layers is 
+supposed to be 𝜇𝑐,1 = 0.2 𝑒𝑣,  𝜇𝑐,2 = 0.3 𝑒𝑣. The thickness of the SiO2 layer is 150 nm. The incident angle is 𝜃 =
+450.   
+ 
+Figure. 7. The reflected group delay versus frequency for different values of SiO2 thickness. It is supposed that both 
+graphene layers have similar thicknesses (𝛥1 = 𝛥2 = 𝛥 = 0.33 𝑛𝑚).  The chemical potential of graphene layers is 
+supposed to be 𝜇𝑐,1 = 0.2 𝑒𝑣,  𝜇𝑐,2 = 0.3 𝑒𝑣. The thickness of the hBN layer is 100 nm. The incident angle is 𝜃 =
+450.   
+ 
+4.  Conclusion                        
+In this article, we studied the characteristics of the reflected beam from graphene-based hBN heterostructure. 
+Analytical expressions were obtained for calculating the reflection characteristics. A large value of the reflected group 
+delay was seen in the lower Reststrahlen band; therefore, this frequency range was chosen to be studied. To show the 
+tunability of the proposed structure, the variations of the reflected beam as a function of frequency were depicted and 
+investigated for various values of chemical potential. Our results reported a large value of the reflected group delay, 
+i.e. 𝜏𝑟 = 15.3 𝑝𝑠, at the frequency of 24.9 THz. Moreover, we showed that the thickness of graphene sheets, the hBN 
+medium, and the SiO2 layer can change the quality of the reflected beam more effectively. The authors believe that 
+the presented study can be utilized for the design of optical delay structures in the mid-infrared region.                           
+ 
+      
+Declarations 
+Ethics Approval: Not Applicable. 
+Consent to Participate: Not Applicable. 
+Consent for Publication: Not Applicable. 
+Funding: The authors received no specific funding for this work.  
+Conflicts of Interest/ Competing Interests: The authors declare no competing interests.  
+
+9 
+ 
+Availability of Data and Materials: Not Applicable.      
+Code availability: Not Applicable.   
+Authors' Contributions: M. B. Heydari proposed the main idea of this work and performed the analytical modeling. 
+M. Karimipour conducted the numerical simulations and wrote the manuscript. M. Mohammadi Shirkolaei analyzed 
+the results and reviewed the paper.       
+ 
+References         
+[1] 
+P. Tassin, T. Koschny, and C. M. Soukoulis, "Graphene for terahertz applications," Science, vol. 341, pp. 
+620-621, 2013. 
+[2] 
+M. Zhai, H. Peng, X. Wang, X. Wang, Z. Chen, and W. Yin, "The Conformal HIE-FDTD Method for 
+Simulating Tunable Graphene-Based Couplers for THz Applications," IEEE Transactions on Terahertz 
+Science and Technology, vol. 5, pp. 368-376, 2015. 
+[3] 
+M. B. Heydari and M. H. V. Samiei, "Graphene-Based Couplers: A Brief Review," arXiv preprint 
+arXiv:2010.09462, 2020. 
+[4] 
+F. Xu, H. Zhang, and Y. Sun, "Compact graphene directional couplers based on dielectric ridges for planar 
+integration," Optik-International Journal for Light and Electron Optics, vol. 131, pp. 588-591, 2017. 
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+D. Correas-Serrano, J. S. Gomez-Diaz, J. Perruisseau-Carrier, and A. Álvarez-Melcón, "Graphene-Based 
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+

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+arXiv:2301.03807v1  [math.RA]  10 Jan 2023
+UNIVERSAL CONSTRUCTIONS FOR POISSON ALGEBRAS.
+APPLICATIONS
+A. L. AGORE AND G. MILITARU
+Abstract. We introduce the universal algebra of two Poisson algebras P and Q as
+a commutative algebra A := P(P, Q) satisfying a certain universal property.
+The
+universal algebra is shown to exist for any finite dimensional Poisson algebra P and
+several of its applications are highlighted. For any Poisson P-module U, we construct a
+functor U⊗−: AM → QPM from the category of A-modules to the category of Poisson
+Q-modules which has a left adjoint whenever U is finite dimensional. Similarly, if V is
+an A-module, then there exists another functor −⊗V : P PM → QPM connecting the
+categories of Poisson representations of P and Q and the latter functor also admits a
+left adjoint if V is finite dimensional. If P is n-dimensional, then P(P) := P(P, P) is
+the initial object in the category of all commutative bialgebras coacting on P. As an
+algebra, P(P) can be deescribed as the quotient of the polynomial algebra k[Xij | i, j =
+1, · · · , n] through an ideal generated by 2n3 non-homogeneous polynomials of degree
+≤ 2. Two applications are provided. The first one describes the automorphisms group
+AutPoiss(P) as the group of all invertible group-like elements of the finite dual P(P)o.
+Secondly, we show that for an abelian group G, all G-gradings on P can be explicitly
+described and classified in terms of the universal coacting bialgebra P(P).
+Introduction
+Introduced in Hamiltonian mechanics as the dual of the category of classical mechanical
+systems, Poisson algebras play an important role in the study of quantum groups, dif-
+ferential geometry, noncommutative geometry, integrable systems, quantum field theory
+or vertex operator algebras (see [12, 13, 15, 18, 19, 20, 30]). Poisson algebras can be
+thoughth of as the algebraic counterpart of Poisson manifolds which are smooth man-
+ifolds M whose commutative algebra C∞(M, R) of real smooth functions is endowed
+with a Lie bracket [−, −] satisfying the Leibniz rule, i.e. C∞(M, R) is a Poisson algebra.
+In this paper we introduce and study some universal objects for Poisson algebras and
+highlight their main applications having as sourse of inspiration the previous work of
+Sweedler [28], Manin [22] and Tambara [25] for Hopf algebra (co)actions on associative
+algebras. From a categorical point of view, the existence of universal objects with a
+certain property, for a given category C can shed some light on the structure of the
+category C itself. In particular, the existence and description of universal objects (groups
+2020 Mathematics Subject Classification. 17B63, 16T05, 16T10, 16W20, 16W50.
+Key words and phrases. Poisson algebras, universal constructions, automorphisms group, gradings.
+This work was supported by a grant of the Ministry of Research, Innovation and Digitization,
+CNCS/CCCDI – UEFISCDI, project number PN-III-P4-ID-PCE-2020-0458, within PNCDI III.
+1
+
+2
+A. L. AGORE AND G. MILITARU
+or ”group like objects” such as Lie groups, algebraic groups, Hopf algebras, groupoids or
+quantum groupoids, etc.) which act or coact on a fixed object O in a certain category
+C has often various applications in many areas of mathematics.
+An elementary but
+illuminating example is the following: let O be a given object in a certain category
+C and consider the category ActGrO of all groups that act on O, i.e. the objects in
+ActGrO are pairs (G, ϕ) consisting of a discrete group G and a morphism of groups
+ϕ : G → AutC(O), where AutC(O) denotes the automorphisms group of the object O
+in C. Then the category ActGrO has a final object, namely
+�
+AutC(O), Id
+�
+. Now, if we
+replace the discrete groups that act on the fixed object O in C, by some other ”groups like
+objects” from a certain more sophisticated category D (for instance, Lie groups, algebraic
+groups, Hopf algebras, etc.) which (co)act on O and if moreover we ask the (co)action
+to preserve the algebraic, differential or topological structures which might exist on O,
+then things become very complicated.
+Indeed, the first obstacle we encounter is the
+fact that AutC(O) might not be an object inside the category D anymore. However,
+even in this complicated situation, it is possible for the above result to remain valid but,
+however, the construction of the final object will be far more complicated. Furthermore,
+it is to expect that, if it exists, this final object will contain important information on
+the entire automorphisms group of the object O. To the best of our knowledge, the first
+result in this direction was proved by Sweelder [28, Theorem 7.0.4] in the case where C
+is the category of associative algebras and D is the category of bialgebras: if A is a fixed
+associative algebra then the category of all bialgebras H that act on A (i.e. A is an
+H-module algebra) has a final object M(A), called by Sweedler the universal measuring
+bialgebra of A. The dual situation of coactions of bialgebras on a fixed algebra A, was first
+considered in the case when C is the category of graded algebras by Manin [22] for reasons
+related to non-commutative geometry, and in the general case by Tambara [25]. If A is an
+associative algebra, necessarily finite dimensional this time around, then the category of
+all bialgebras that coact on A (i.e. A is an H-comodule algebra) has an intial object a(A).
+The results have been extended in recent years to the setting of bialgebroids coactions in
+[7, 8]. Furthermore, the usual automorphisms group AutAlg(A) of A is indeed recovered
+as the group of all invertible group-like elements of the finite dual a(A)o [23, Theorem
+2.1] and a(A)o is just Sweedler’s final object in the category of all bialgebras that act
+on A [25, Remark 1.3]. The two results above remains valid if we take the category of
+Hopf algebras instead of bialgebras: in particular, the Hopf envelope of a(A), denoted by
+aut(A), is called in non-commutative geometry the non-commutative symmetry group of
+A [27] and its description is a very complicated matter. The existence and description of
+these universal (co)acting bialgebras/Hopf algebras has been considered recently in [2] in
+the context of Ω-algebras. The duality between Sweedler’s and Manin-Tambara’s objects
+has been extended to this general setting and necessary and sufficient conditions for the
+existence of the universal coacting bialgebras/Hopf algebras, which roughly explains the
+need for assuming finite-dimensionality in Manin-Tambara’s constructions, are given.
+Furthermore, universal coacting objects for Poisson algebras have also been considered
+in [3] but from a different perspective, leading to entirely different constructions. We
+only point out that in [3], the universal coacting object considered is actually a Poisson
+Hopf algebra. For more background on the importance and the applications of universal
+
+UNIVERSAL CONSTRUCTIONS FOR POISSON ALGEBRAS
+3
+bialgebras/Hopf algebras in various areas of mathematics we refer to [4, 7, 8, 10, 11, 16,
+17].
+The paper is organized as follows. Definition 2.1 introduces the key object of our work,
+namely the universal algebra of two Poisson algebras P and Q, as a pair
+�
+P(P, Q), η
+�
+consisting of a commutative algebra A := P(P, Q) and a Poisson algebra homomorphism
+η: Q → P ⊗ P(P, Q) satisfying a certain universal property. Theorem 2.2 proves that
+if P is finite dimensional, then the universal algebra P(P, Q) of P and Q exists and
+its explicit construction is provided. This result has two important consequences: as
+proved in Theorem 2.5, for a fixed Poisson P-module U there exists a canonical functor
+U ⊗ −: AM → QPM from the category of usual A-modules (i.e. representations of
+the associative algebra A) to the category of Poisson Q-modules (i.e. Poisson repre-
+sentations of Q) and moreover, if U is finite dimensional this functor has a left adjoint
+(Theorem 2.6). Secondly, if V is an A-module, then there exists a canonical functor
+− ⊗ V : P PM → QPM connecting the categories of Poisson modules over P and Q
+and, furthermore, if V is finite dimensional then the aforementioned functor has a left
+adjoint (Theorem 2.7). These results provide answers, at the level of Poisson algebras,
+to the following general question: if O1 and O2 are two mathematical objects (not nec-
+essary in the same category), is it possible to construct ”canonical functors” between the
+representation categories Rep(O1) and Rep(O2) of the two objects?
+In Section 3 we consider three more applications of our constructions.
+For an n-
+dimensional Poisson algebra P, we denote P(P) := P(P, P) and we construct P(P)
+as the quotient of the polynomial algebra k[Xij | i, j = 1, · · · , n] through an ideal gen-
+erated by 2n3 non-homogeneous polynomials of degree ≤ 2. P(P) has a canonical bial-
+gebra structure and Theorem 3.3 shows that P(P) is the initial object of the category
+CoactBialgP of all commutative bialgebras coacting on P and, for this reason, we call it
+the universal coacting bialgebra of P. As in the case of Lie [5] or associative algebras [23],
+the universal bialgebra P(P) has two important applications, which provide the theoret-
+ical answer for Poisson algebras, of the following open questions: (1) Describe explicitly
+the automorphisms group of a given Poisson algebra P; (2) Describe and classify all
+G-gradings on P for a given abelian group G. More precisely, Theorem 3.6 proves that
+there exists an isomorphism of groups between the group of all Poisson automorphisms
+of P and the group of all invertible group-like elements of the finite dual P(P)o. The
+second application is given in Theorem 3.9: for an abelian group G, all G-gradings on
+a finite dimensional Poisson algebra P are described and classified in terms of bialgebra
+homomorphisms P(P) → k[G]. By taking Takeuchi’s commutative Hopf envelope of
+P(P), we obtain that the category CoactHopfP of all commutative Hopf algebras coact-
+ing on P has an initial object H(P) (Corollary 3.4). It is reasonable to hope that H(P)
+will play the role of a non-commutative symmetry group of the Poisson algebra P. This
+expectation is based on the fact that the concept of Poisson H-comodule algebra which
+we are dealing with, is the algebraic counterpart of the action of an algebraic groups on
+an affine Poisson variety [14, Example 2.20].
+
+4
+A. L. AGORE AND G. MILITARU
+1. Preliminaries
+All vector spaces, (bi)linear maps, unadorned tensor products, associative, Lie or Pois-
+son algebras and so on are over an arbitrary field k. Throughout, δs,1 will stand for
+Kronecker’s symbol. A Poisson algebra is a vector space P which admits both an (non-
+necessarily unital) associative commutative algebra and a Lie algebra such that for all
+x, y, z ∈ P we have:
+[x, yz] = [x, y] z + y [x, z].
+(1)
+A morphism of two Poisson algebras P1 and P2 is a linear map f : P1 → P2 which is
+both an algebra homomorphism as well as a Lie algebra homomorphism; if P1 and P2
+are unital Poisson algebras then a Poisson homomorphism will be assumed to preserve
+units. We denote by AutPoiss(P) the automorphisms group of a Poisson algebra P.
+Let P be a Poisson algebra.
+A (left) Poisson P-module [6, 29] is a vector space V
+equipped with two bilinear maps ⊲: P × V → V and ⇀: P × V → V such that (V, ⊲) is
+a left P-module, (V, ⇀) is a left Lie P-module satisfying the following two compatibility
+conditions for all a, b ∈ P and x ∈ V :
+(ab) ⇀ x = a ⊲ (b ⇀ x) + b ⊲ (a ⇀ x)
+(2)
+[a, b] ⊲ x = a ⇀ (b ⊲ x) − b ⇀ (a ⊲ x)
+(3)
+We denote by PPM the category of Poisson P-modules having as morphisms all linear
+maps which are compatible with both actions.
+Remarks 1.1. 1. The category PPM of Poisson P-modules is equivalent to the category
+of usual left P e-modules ([29, Corollary 1]), where P e is the universal enveloping algebra
+of P as constructed there. In particular, for any set S we denote by (P e)(S), the free
+P e-module generated by S, which is the free Poisson P-module generated by S. Any
+quotient (P e)(S)/N through a Poisson submodule N generated by a system of generators
+R is called the free Poisson P-module generated by S and the relations R.
+2. A representation of a Poisson algebra P on a vector space V [6, Remarks 2.9] is a pair
+(ψ, ϕ) consisting of an algebra map ψ : P → Endk(V ), a Lie algebra map ϕ : P → glk(V )
+such that for any a, b ∈ P:
+ϕ(ab) = ψ(a) ◦ ϕ(b) + ψ(b) ◦ ϕ(a),
+ψ([a, b]) = ϕ(a) ◦ ψ(b) − ϕ(b) ◦ ψ(a).
+The concepts of a Poisson P-module structure on V and a representation of P on V are
+obviously equivalent.
+We shall denote by Poissk, Poiss1
+k and ComAlgk the categories of Poisson, unital Poisson
+and respectively unital commutative associative algebras over k. Furthermore, the cat-
+egory of commutative bialgebras (resp. Hopf algebras) is denoted by ComBiAlgk (resp.
+ComHopfk). For a coalgebra C we denote by G(C) the set of group like elements of
+C, i.e. G(C) := {x ∈ C | ∆(x) = x ⊗ x and ε(x) = 1}. If B is a bialgebra, then G(B)
+is a monoid with respect to the multiplication on B. Throughout, for a bialgebra B,
+we denote by Bo its finite dual. Recall that if H and L are two bialgebras then the
+abelian group Homk (H, L) is an associative algebra under the convolution product [28]:
+(θ1 ⋆ θ2)(h) := � θ1(h(1))θ2(h(2)), for all θ1, θ2 ∈ Homk (H, L) and h ∈ H.
+
+UNIVERSAL CONSTRUCTIONS FOR POISSON ALGEBRAS
+5
+If H is a commutative bialgebra (or a Hopf algebra), then a Poisson algebra P is called
+a right Poisson H-comodule algebra [9] (we also say that H coacts on P) if there exists
+ρP : P → P ⊗ H a Poisson algebra map (the Poisson algebra structures on P ⊗ H
+are given by (4) below) that is also a right H-comodule stucture on P. If (P, ρP ) is a
+right Poisson H-comodule algebra, then the subalgebra of coinvariants P co(H) := {p ∈
+P | ρP (p) = p ⊗ 1H} is a Poisson subalgebra of P. For a fixed Poisson algebra P we
+denote by CoactBialgP (resp. CoactHopfP ) the category of all commutative bialgebras
+(resp. Hopf algebras) coacting on P. That is, the objects are all pairs (H, ρP ) consisting
+of a commutative bialgebra (resp. Hopf algebra) H together with a structure of a right
+Poisson H-comodule algebra ρP : P → P ⊗ H while morphisms f : (H, ρP ) → (H′, ρ′
+P )
+in CoactBialgP are bialgebra maps f : H → H′ such that (IdP ⊗ f) ◦ ρP = ρ′
+P .
+Examples 1.2. 1. The first basic example of a Poisson H-comodule algebra is the one
+induced by G-graded Poisson algebras. Recall that, given an abelian group G and a
+Poisson algebra P, a G-grading on P is a vector space decomposition P = ⊕σ∈G Pσ such
+that PσPτ ⊆ Pστ and [Pσ, Pτ] ⊆ Pστ, for all σ, τ ∈ G. Two G-gradings P = ⊕σ∈G Pσ =
+⊕σ∈G P
+′
+σ on P are called isomorphic if there exists w ∈ AutPoiss(P) an automorphism
+of P such that w(Pσ) = P
+′
+σ, for all σ ∈ G. Let k[G] be the group algebra of G. By
+extending a well known result in Hopf algebra theory ([26, Excercise 3.2.21]) one can
+easily see that there is a bijection between the set of all right Poisson k[G]-comodule
+structures ρ: P → P ⊗ k[G] on the Poisson algebra P and the set of all G-gradings on
+P = ⊕σ∈G Pσ. The bijection is given such that xσ ∈ Pσ if and only if ρ(xσ) = xσ ⊗ σ,
+for all σ ∈ G.
+2. The second example of a Poisson comodule algebra comes from algebraic geometry
+[14, Example 2.20]: if V is an affine Poisson variety (i.e. the coordinate ring k[V ] of
+V is a Poisson algebra) and G is an algebraic group acting on V via automorphisms of
+Poisson varieties, then k[V ] is a Poisson k[G]-comodule algebra.
+For further details concerning the study of Poisson algebras see [12, 20] and the references
+therein and for undefined concepts on category theory (resp. Hopf algebras) we refer the
+reader to [21] (resp. [26, 28]).
+2. The universal algebra of two Poisson algebras
+Before introducing the main characters of this paper we make the following key obser-
+vation: if P is a Poisson algebra and A is a commutative associative algebra then P ⊗ A
+is a Poisson algebra. The associative algebra structure and the Lie bracket are defined
+as follows for all x, y ∈ P and a, b ∈ A:
+(x ⊗ a) (y ⊗ b) = xy ⊗ ab,
+[x ⊗ a, y ⊗ b] = [x, y] ⊗ ab.
+(4)
+
+6
+A. L. AGORE AND G. MILITARU
+Indeed, having in mind that A is a commutative associative algebra, we have:
+�
+x ⊗ a, (y ⊗ b)(z ⊗ c)
+� (4)
+= [x ⊗ a, yz ⊗ bc]
+(4)
+= [x, yz] ⊗ abc
+(1)
+= [x, y] z ⊗ abc + y [x, z] ⊗ abc
+(4)
+= ([x, y] ⊗ ab)(z ⊗ c) + (y ⊗ b)([x, z] ⊗ ac)
+(4)
+= [x ⊗ a, y ⊗ b] (z ⊗ c) + (y ⊗ b) [x ⊗ a, z ⊗ c]
+for all x, y, z ∈ P and a, b, c ∈ A, i.e. (1) holds for P ⊗ A. Furthermore, if f : A → B
+is an algebra map then IdP ⊗ f : P ⊗ A → P ⊗ B is a morphism of Poisson algebras.
+To conclude, given a Poisson algebra P, assigning A �→ P ⊗ A defines a functor P ⊗ − :
+ComAlgk → Poissk from the category of commutative algebras to the category of Poisson
+algebras. With this remark in hand we can now introduce the following concept:
+Definition 2.1. Let P and Q be two Poisson algebras. The universal algebra of P and
+Q is a pair
+�
+P(P, Q), η
+�
+consisting of a commutative algebra P(P, Q) ∈ ComAlgk and
+a Poisson algebra homomorphism η: Q → P ⊗ P(P, Q) satisfying the following univer-
+sal property: for any commutative algebra A and any Poisson algebra homomorphism
+g: Q → P ⊗ A there exists a unique algebra homomorphism θ: P(P, Q) → A such that
+the following diagram is commutative:
+Q
+η
+�
+g
+�❑
+❑
+❑
+❑
+❑
+❑
+❑
+❑
+❑
+❑
+❑
+P ⊗ P(P, Q)
+IdP ⊗θ
+�
+P ⊗ A
+i.e. g =
+�
+IdP ⊗ θ
+�
+◦ η
+(5)
+If Q = P then P(P) := P(P, P) will be called the universal coacting bialgebra on P 1.
+The universal algebra of two Poisson algebras P and Q, if exists, it is unique up to
+an isomorphism of algebras. In what follows we prove that if P is a finite dimensional
+Poisson algebra and Q an arbitrary Poisson algebra, then the universal algebra P(P, Q)
+of P and Q exists and we will provide its explicit construction. We formulate this result
+in terms of adjoint functors, as the Poisson algebra version of [25, Theorem 1.1].
+Theorem 2.2. Let P be a finite dimensional Poisson algebra. Then the functor P ⊗
+− : ComAlgk → Poissk has a left adjoint P(P, −) : Poissk → ComAlgk. Furthermore, if
+Q is an arbitrary Poisson algebra, then P(P, Q) is the universal algebra of P and Q.
+Proof. Let n ∈ N∗ be a positive integer and {e1, · · · , en} a basis of the Poisson algebra P.
+We denote by {τ s
+i,j | i, j, s = 1, · · · , n} and {µs
+i,j | i, j, s = 1, · · · , n} the structure constants
+of P with respect to the associative and Lie structures, i.e. for all i, j = 1, · · · , n we
+have:
+ei ej =
+n
+�
+s=1
+τ s
+i,j es,
+[ei, ej]P =
+n
+�
+s=1
+µs
+i,j es.
+(6)
+We will construct explicitly a left adjoint P(P, −) : Poissk → ComAlgk for the tensor
+product functor P ⊗ − : ComAlgk → Poissk. To this end, let Q be a Poisson algebra
+1The terminology is explained by Theorem 3.3 below.
+
+UNIVERSAL CONSTRUCTIONS FOR POISSON ALGEBRAS
+7
+and consider {fi | i ∈ I} to be its basis. Then, for all i, j ∈ I, we can find two finite
+subsets Ui,j and Vi,j of I such that:
+fi fj =
+�
+u∈Ui,j
+αu
+i,j fu,
+[fi, fj]Q =
+�
+u∈Vi,j
+βu
+i,j fu
+(7)
+for some scalars αu
+i,j, βu
+i,j ∈ k. Consider now k[Xsi | s = 1, · · · , n, i ∈ I] to be the usual
+polynomial algebra and let
+P(P, Q) := k[Xsi |s = 1, · · · , n, i ∈ I]/J
+where J is the ideal generated by all polynomials of the form:
+Γ(P, Q)
+(a,i,j) =
+�
+u∈Ui,j
+αu
+i,j Xau −
+n
+�
+s,t=1
+τ a
+s,t XsiXtj
+(8)
+Ω(P, Q)
+(a,i,j) =
+�
+u∈Vi,j
+βu
+i,j Xau −
+n
+�
+s,t=1
+µa
+s,t XsiXtj
+(9)
+for all a = 1, · · · , n and i, j ∈ I.
+Denoting xsi := �
+Xsi, where �
+Xsi stands for the
+equivalence class of Xsi in the quotient algebra P(P, Q), it follows that the relations
+below hold in P(P, Q):
+�
+u∈Ui,j
+αu
+i,j xau =
+n
+�
+s,t=1
+τ a
+s,t xsixtj
+(10)
+�
+u∈Vi,j
+βu
+i,j xau =
+n
+�
+s,t=1
+µa
+s,t xsixtj
+(11)
+for all a = 1, · · · , n and i, j ∈ I. Next, we consider the following linear map:
+ηQ : Q → P ⊗ P(P, Q),
+ηQ(fi) :=
+n
+�
+s=1
+es ⊗ xsi,
+for all i ∈ I.
+(12)
+We will see that ηQ is in fact a Poisson algebra map; indeed, for all i, j ∈ I we have:
+[ηQ(fi), ηQ(fj)]P ⊗P(P, Q) =
+� n
+�
+s=1
+es ⊗ xsi,
+n
+�
+t=1
+et ⊗ xtj
+�
+P ⊗P(P, Q)
+=
+n
+�
+s,t=1
+[es, et]P ⊗ xsixtj =
+n
+�
+a=1
+ea ⊗
+�
+n
+�
+s, t=1
+µa
+s,t xsixtj
+�
+(11)
+=
+n
+�
+a=1
+ea ⊗
+� �
+u∈Vi,j
+βu
+i,j xau
+�
+=
+�
+u∈Vi,j
+βu
+i,j ηQ(fu) = ηQ([fi, fj]Q)
+
+8
+A. L. AGORE AND G. MILITARU
+and
+ηQ(fi) ηQ(fj) =
+� n
+�
+s=1
+es ⊗ xsi
+� � n
+�
+t=1
+et ⊗ xtj
+�
+=
+n
+�
+s,t=1
+es et ⊗ xsixtj
+=
+n
+�
+a=1
+ea ⊗
+�
+n
+�
+s, t=1
+τ a
+s,t xsixtj
+�
+(10)
+=
+n
+�
+a=1
+ea ⊗
+� �
+u∈Ui,j
+αu
+i,j xau
+�
+=
+�
+u∈Ui,j
+αu
+i,j ηQ(fu)
+= ηQ(fi fj)
+This shows that ηQ is indeed a Poisson algebra homomorphism, as claimed. The next
+step of the proof consists in showing that for any Poisson algebra Q and any commutative
+algebra A the map defined below is bijective:
+γQ, A : HomAlgk (P(P, Q), A) → HomPoissk (Q, P ⊗ A),
+γQ, A(θ) =
+�
+IdP ⊗ θ
+�
+◦ηQ (13)
+To this end, let g: Q → P ⊗ A be a Poisson algebra homomorphism.
+We have to
+prove that there exists a unique algebra homomorphism θ: P(P, Q) → A such that
+g =
+�
+IdP ⊗ θ
+�
+◦ ηQ. Let {dsi | s = 1, · · · , n, i ∈ I} be a family of elements of A such that
+for all i ∈ I we have:
+g(fi) =
+n
+�
+s=1
+es ⊗ dsi
+(14)
+Furthermore, as g: Q → P ⊗ A is a Poisson algebra map, we can easily conclude that
+the following compatibilities hold for all a = 1, · · · , n and i, j ∈ I:
+�
+u∈Ui,j
+αu
+i,j dau =
+n
+�
+s,t=1
+τ a
+s,t dsidtj
+(15)
+�
+u∈Vi,j
+βu
+i,j dau =
+n
+�
+s,t=1
+µa
+s,t dsidtj
+(16)
+The universal property of the polynomial algebra yields a unique algebra homomorphism
+v: k[Xsi |s = 1, · · · , n, i ∈ I] → A such that v(Xsi) = dsi, for all s = 1, · · · , n and i ∈ I.
+Furthermore, we have J ⊆ Ker(v), where J is the ideal generated by all polynomials
+listed in (8) and (9). Indeed, for all i, j ∈ I and a = 1, · · · , n we have:
+v
+�
+Γ(P, Q)
+(a,i,j)
+�
+= v
+� �
+u∈Ui,j
+αu
+i,j Xau −
+n
+�
+s,t=1
+τ a
+s,t XsiXtj
+�
+=
+�
+u∈Ui,j
+αu
+i,j dau −
+n
+�
+s,t=1
+τ a
+s,t dsidtj
+(15)
+= 0
+v
+�
+Ω(P, Q)
+(a,i,j)
+�
+= v
+� �
+u∈Vi,j
+βu
+i,j Xau −
+n
+�
+s,t=1
+µa
+s,t XsiXtj
+�
+=
+�
+u∈Vi,j
+βu
+i,j dau −
+n
+�
+s,t=1
+µa
+s,t dsidtj
+(16)
+= 0
+Thus, there exists a unique algebra homomorphism θ: P(P, Q) → A such that θ(xsi) =
+dsi, for all s = 1, · · · , n and i ∈ I. We are left to show that g =
+�
+IdP ⊗ θ
+�
+◦ ηQ. To this
+end, for all i ∈ I we have:
+�
+IdP ⊗ θ
+�
+◦ ηQ(fi) =
+�
+IdP ⊗ θ
+�� n
+�
+s=1
+es ⊗ xsi
+�
+=
+n
+�
+s=1
+es ⊗ dsi
+(30)
+= g(fi),
+
+UNIVERSAL CONSTRUCTIONS FOR POISSON ALGEBRAS
+9
+as desired.
+We are left to show that θ is the unique morphism with this property.
+Indeed, consider ˜θ: P(P, Q) → A to be another algebra homomorphism such that
+�
+IdP ⊗
+˜θ
+�
+◦ ηQ(fi) = g(fi), for all i ∈ I. Then, �n
+s=1 es ⊗ ˜θ(xsi) = �n
+s=1 es ⊗ dsi, and hence
+˜θ(xsi) = dsi = θ(xsi), for all s = 1, · · · , n and i ∈ I. As the set {xsi |s = 1, · · · , n, i ∈ I }
+generates the algebra P(P, Q) we can conclude that ˜θ = θ. To summarize, we proved
+that the map γQ, A given by (13) is bijective.
+The only thing left to show is that given a finite dimensional Poisson algebra P, assigning
+Q �→ P(P, Q) defines a functor P(P, −): Poissk → ComAlgk. Indeed, let u: Q1 → Q2
+be a Poisson algebra homomorphism. Applying the bijectivity of the map defined by (13)
+for the Poisson algebra homomorphism ηQ2 ◦ u, yields a unique algebra homomorphism
+θ: P(P, Q1) → P(P, Q2) such that:
+�
+IdP ⊗ θ
+�
+◦ ηQ1 = ηQ2 ◦ u
+(17)
+By considering P(P, u) to be this unique morphism θ, the functor P(P, −) is fully
+defined. Moreover, it can now be easily checked that P(P, −) is indeed a functor and
+that γQ, A is natural in both variables. Therefore, the functor P(P, −) is the left adjoint
+of the functor P ⊗ −.
+Finally, the bijectivity of the map (13) shows that the pair
+�
+P(P, Q), ηQ
+�
+is indeed the universal algebra of P and Q.
+□
+Remark 2.3. Theorem 2.2 remains valid if we replace Poissk by the category Poiss1
+k
+of unital Poisson algebras. If P is a unital finite dimensional Poisson algebra, then the
+functor P ⊗ − : ComAlgk → Poiss1
+k has a left adjoint P1(P, −): Poiss1
+k → ComAlgk
+which is constructed as follows. If {e1, · · · , en} is a basis of the Poisson algebra P such
+that e1 := 1P and Q is a unital Poisson algebra with basis {fi | i ∈ I} such that fi0 := 1Q
+then we define
+P1(P, Q) := P(P, Q)/L
+where L is the ideal of P(P, Q) generated by xsi0 − δs,1, for all s = 1, · · · , n. These new
+relations are necessary and sufficient for the map ηQ: Q → P ⊗ P1(P, Q) defined in (12)
+to be unital, i.e. ηQ(1Q) = 1P ⊗1. The rest of the proof goes exactly as for Theorem 2.2.
+Furthermore, Theorem 2.2 can be generalized to the category of Jacobi algebras by
+repeating verbatim the above proof.
+Recall that a Jacobi algebra [6] is a quadruple
+J = (J, mJ, 1J, [−, −]), where (J, mJ, 1J) is a unital commutative algebra, (A, [−, −])
+is a Lie algebra such that for all a, b, c ∈ J we have:
+[ab, c] = a [b, c] + [a, c] b − ab [1A, c]
+(18)
+We can prove that for any Jacobi algebra J and any commutative algebra A, the tensor
+product J ⊗ A is a Jacobi algebra with the structures given by (4). If we denote by Jack
+the category of Jacobi algebras, then for any finite diminesional Jacobi algebra J, the
+functor J ⊗ − : ComAlgk → Jack has a left adjoint.
+The universal algebra P(P, Q) of two Poisson algebras P and Q as constructed in Theo-
+rem 2.2 is an important tool for comparing the two Poisson algebras: the first application
+shows that the set of all usual algebra maps P(P, Q) → k parameterize the space of all
+Poisson algebra maps Q → P. Indeed, by considering A := k, the bijection described in
+(13) comes down to the following:
+
+10
+A. L. AGORE AND G. MILITARU
+Corollary 2.4. Let P and Q be two Poisson algebras such that P is finite dimensional.
+Then the following map is bijective:
+γ : HomAlgk (P(P, Q), k) → HomPoissk (Q, P),
+γ(θ) :=
+�
+IdP ⊗ θ
+�
+◦ηQ
+(19)
+The next applications of the universal algebra P(P, Q) are more nuanced and refer
+to representations (i.e. Poisson modules) of the two Poisson algebras P and Q. In the
+sequel, we will use the explicit description through generators and relations of the algebra
+P(P, Q) provided in the proof of Theorem 2.2.
+Theorem 2.5. Let P and Q be Poisson algebras such that P is finite dimensional,
+A = P(P, Q) the corresponding universal algebra, (U, ◮, ↷) ∈ PPM a Poisson P-
+module and (V, ·) ∈ AM an A-module.
+Then (U ⊗ V, ⊲, ⇀) ∈ QPM is a Poisson Q-module where the actions of Q on U ⊗ V
+are given for all i ∈ I, l ∈ U and t ∈ V by:
+fi ⊲ (l ⊗ t) =
+n
+�
+j=1
+(ej ◮ l) ⊗ (xji · t)
+(20)
+fi ⇀ (l ⊗ t) =
+n
+�
+j=1
+(ej ↷ l) ⊗ (xji · t)
+(21)
+In particular, any fixed (U, ◮, ↷) ∈ P PM yields a functor U ⊗ −: AM → QPM from
+the category of A-modules to the category of Poisson Q-modules; similarly, any fixed
+(V, ·) ∈ AM gives rise to a functor − ⊗ V : PPM → QPM connecting the categories of
+Poisson modules over P and Q.
+Proof. We start by showing that (U ⊗ V, ⊲) is a left Q-module. To thie end, we have:
+(fifj) ⊲ (l ⊗ t)
+(7)
+=
+�
+u∈Ui,j
+αu
+i,jfu ⊲ (l ⊗ t)
+(20)
+=
+�
+u∈Ui,j,r=1,n
+(αu
+i,jer ◮ l) ⊗ (xru · t)
+=
+n
+�
+r=1
+(er ◮ l) ⊗
+� �
+u∈Ui,j
+αu
+i,jxru
+�
+· t
+(10)
+=
+n
+�
+r,s,p=1
+τ r
+s,p (er ◮ l) ⊗ (xsixpj) · t
+=
+n
+�
+s,p=1
+� n
+�
+r=1
+τ r
+s,p er
+�
+◮ l ⊗ (xsixpj) · t
+(6)
+=
+n
+�
+s,p=1
+(esep) ◮ l ⊗ (xsixpj) · t
+=
+n
+�
+s,p=1
+es ◮ (ep ◮ l) ⊗ (xsixpj) · t =
+n
+�
+p=1
+� n
+�
+s=1
+es ◮ (ep ◮ l) ⊗ xsi · (xpj · t)
+�
+(20)
+= fi ⊲
+n
+�
+p=1
+ep ◮ l ⊗ xpj · t
+(20)
+= fi ⊲
+�
+fj ⊲ (l ⊗ t)
+�
+We point out that (U ⊗V, ⇀) being a left Lie Q-module can be proved exactly as in (the
+proof of) [1, Theorem 2.1]. The proof will be finished once we prove that compatibilities
+
+UNIVERSAL CONSTRUCTIONS FOR POISSON ALGEBRAS
+11
+(2) and (3) hold for (U ⊗ V, ⊲, ⇀).
+Indeed, as compatibilities (2) and (3) hold for
+(U, ◮, ↷) and A is a commutative algebra, for all i, j ∈ I and l ∈ U, t ∈ V , we have:
+(fifj) ⇀ (l ⊗ t)
+(7)
+=
+�
+u∈Ui,j
+αu
+i,jfu ⇀ (l ⊗ t)
+(21)
+=
+�
+u∈Ui,j,r=1,n
+(αu
+i,jer ↷ l) ⊗ (xru · t)
+=
+n
+�
+r=1
+(er ↷ l) ⊗
+� �
+u∈Ui,j
+αu
+i,jxru
+�
+· t
+(10)
+=
+n
+�
+r,s,p=1
+τ r
+s,p (er ↷ l) ⊗ (xsixpj) · t
+=
+n
+�
+s,p=1
+� n
+�
+r=1
+τ r
+s,p er
+�
+↷ l ⊗ (xsixpj) · t
+(6)
+=
+n
+�
+s,p=1
+(esep) ↷ l ⊗ (xsixpj) · t
+(2)
+=
+n
+�
+s,p=1
+�
+es ◮ (ep ↷ l) + ep ◮ (es ↷ l)
+�
+⊗(xsixpj) · t
+=
+n
+�
+s,p=1
+es ◮ (ep ↷ l) ⊗ xsi · (xpj · t) +
+n
+�
+s,p=1
+ep ◮ (es ↷ l) ⊗ xpj · (xsi · t)
+(20)
+= fi ⊲
+n
+�
+p=1
+(ep ↷ l) ⊗ (xpj · t) + fj ⊲
+n
+�
+s=1
+(es ↷ l) ⊗ (xsi · t)
+(21)
+= fi ⊲
+�
+fj ⇀ (l ⊗ t)
+�
++ fj ⊲
+�
+fi ⇀ (l ⊗ t)
+�
+and
+[fi, fj] ⊲ (l ⊗ t)
+(7)
+=
+�
+v∈Vi,j
+βu
+i,j fu ⊲ (l ⊗ t)
+(20)
+=
+�
+u∈Vi,j,r=1,n
+βu
+i,j(er ◮ l) ⊗ (xru · t)
+=
+n
+�
+r=1
+(er ◮ l) ⊗
+� �
+u∈Vi,j
+βu
+i,jxru
+�
+· t
+(11)
+=
+n
+�
+r,s,p=1
+µr
+s,p (er ◮ l) ⊗ (xsixpj) · t
+=
+n
+�
+s,p=1
+� n
+�
+r=1
+µr
+s,p er
+�
+◮ l ⊗ (xsixpj) · t
+(6)
+=
+n
+�
+s,p=1
+[es, ep] ◮ l ⊗ (xsixpj) · t
+(3)
+=
+n
+�
+s,p=1
+�
+es ↷ (ep ◮ l) − ep ↷ (es ◮ l)
+�
+⊗ (xsixpj) · t
+=
+n
+�
+s,p=1
+�
+es ↷ (ep ◮ l)
+�
+⊗ xsi · (xpj · t) −
+n
+�
+s,p=1
+�
+ep ↷ (es ◮ l)
+�
+⊗ xpj · (xsi · t)
+(21)
+= fi ⇀
+n
+�
+p=1
+(ep ◮ l) ⊗ (xpj · t) − fj ⇀
+n
+�
+s=1
+(es ◮ l) ⊗ (xsi · t)
+(20)
+= fi ⇀
+�
+fj ⊲ (l ⊗ t)
+�
+− fj ⇀
+�
+fi ⊲ (l ⊗ t)
+�
+which concludes the proof.
+□
+
+12
+A. L. AGORE AND G. MILITARU
+Furhermore, if (U, ◮, ↷) ∈ PPM is finite dimensional then the first functor constructed
+in Theorem 2.5 admits a left adjoint:
+Theorem 2.6. Let P and Q be Poisson algebras such that P is finite dimensional,
+A = P(P, Q) and (U, ◮, ↷) ∈ P PM a finite dimensional Poisson P-module. Then the
+functor U ⊗ −: AM → QPM has a left adjoint U(U, −): QPM → AM.
+Proof. Let {u1, · · · , um}, m ∈ N∗, be a k-basis of the Poisson P-module U and denote
+by γt
+i,j, ωt
+i,j ∈ k the structure constants of U with respect the two module structures, i.e.
+for all i = 1, · · · , n, j = 1, · · · , m we have:
+ei ◮ uj =
+m
+�
+s=1
+γs
+i,j us,
+ei ↷ uj =
+n
+�
+s=1
+ωs
+i,j us
+(22)
+where {e1, · · · , en} is a k-basis of P. The left adjoint U(U, −): QPM → AM of the
+tensor product functor U ⊗ − will be constructed as follows. First, consider (V, ⊢, ↬) ∈
+QPM and {vr | r ∈ J} its k-basis. For all j ∈ I and r ∈ J we can find two finite subsets
+Wj,r and Tj,r of J such that:
+fj ⊢ vr =
+�
+t∈Wj,r
+σt
+j,r vt,
+fj ↬ vr =
+�
+l∈Tj,r
+ηl
+j,r vl
+(23)
+where σt
+j,r, ηl
+j,r ∈ k for all j ∈ I, r ∈ J, t ∈ Wj,r and l ∈ Tj,r (recall that {fi | i ∈ I}
+is a k-basis in Q). Consider now U(U, V ) to be the free A-module generated by the
+set {Yij | i = 1, · · · , m, j ∈ J} and denote by U(U, V ) the quotient of U(U, V ) by its
+A-submodule generated by the following elements:
+�
+p∈Wj,i
+σp
+j,i Ysp −
+m
+�
+t=1
+n
+�
+r=1
+γs
+r,t xrj ⋄ Yti
+(24)
+�
+p∈Tj,i
+ηp
+j,i Ysp −
+m
+�
+t=1
+n
+�
+r=1
+ωs
+r,t xrj ⋄ Yti
+(25)
+for all s = 1, · · · , m, i ∈ J and j ∈ I, where ⋄ denotes the A-module action on U(U, V ).
+Denoting ytj := �
+Ytj, where �
+Ytj stands for the equivalence class of Ytj in the quotient
+module U(U, V ), it follows that the relations below hold in the A-module U(U, V ):
+�
+p∈Wj,i
+σp
+j,i ysp =
+m
+�
+t=1
+n
+�
+r=1
+γs
+r,t xrj ⋄ yti
+(26)
+�
+p∈Tj,i
+ηp
+j,i ysp =
+m
+�
+t=1
+n
+�
+r=1
+ωs
+r,t xrj ⋄ yti
+(27)
+for all s = 1, · · · , m, i ∈ J and j ∈ I. Consider now the following linear map:
+ρV : V → U ⊗ U(U, V ),
+ρV (vr) :=
+m
+�
+s=1
+us ⊗ ysr,
+for all r ∈ J.
+(28)
+
+UNIVERSAL CONSTRUCTIONS FOR POISSON ALGEBRAS
+13
+Note that ρV is a Poisson Q-module map; indeed, for all j ∈ I and i ∈ J we have:
+ρV (fj ⊢ vi)
+(23)
+= ρV
+� �
+p∈Wj,i
+σp
+ji vp
+�
+=
+�
+p∈Wj,i
+m
+�
+s=1
+σp
+ji us ⊗ ysp =
+m
+�
+s=1
+�
+us ⊗
+�
+p∈Wj,i
+σp
+ji ysp
+�
+(26)
+=
+m
+�
+s,t=1
+n
+�
+r=1
+γs
+r,t us ⊗ xrj ⋄ yti =
+m
+�
+t=1
+n
+�
+r=1
+� m
+�
+s=1
+γs
+r,t us
+�
+⊗ xrj ⋄ yti
+(22)
+=
+m
+�
+t=1
+n
+�
+r=1
+er ◮ ut ⊗ xrj ⋄ yti
+(20)
+=
+m
+�
+t=1
+fj ⊲ (ut ⊗ yti) = fj ⊲
+m
+�
+t=1
+ut ⊗ yti
+(28)
+= fj ⊲ ρV (vi)
+and
+ρV (fj ↬ vi)
+(23)
+= ρV
+� �
+p∈Tj,i
+ηp
+ji vp
+�
+=
+�
+p∈Tj,i
+m
+�
+s=1
+ηp
+ji us ⊗ ysp =
+m
+�
+s=1
+�
+us ⊗
+�
+p∈Tj,i
+ηp
+ji ysp
+�
+(27)
+=
+m
+�
+s,t=1
+n
+�
+r=1
+ωs
+r,t us ⊗ xrj ⋄ yti =
+m
+�
+t=1
+n
+�
+r=1
+� m
+�
+s=1
+ωs
+r,t us
+�
+⊗ xrj ⋄ yti
+(22)
+=
+m
+�
+t=1
+n
+�
+r=1
+er ↷ ut ⊗ xrj ⋄ yti
+(21)
+=
+m
+�
+t=1
+fj ⇀ (ut ⊗ yti) = fj ⇀
+m
+�
+t=1
+ut ⊗ yti
+(28)
+= fj ⇀ ρV (vi)
+which concludes our last claim. We can now define for all Poisson Q-modules V and all
+A-modules X, a bijection between HomAM
+�
+U(U, V ), X
+�
+and HomQPM (V, U ⊗ X) as
+follows:
+ΓV,X : HomAM (U(U, V ), X) → HomQPM (V, U ⊗ X), ΓV,X(θ) := (IdU ⊗ θ) ◦ ρV
+(29)
+for all A-module morphisms θ: U(U, V ) → X. To this end, let g: V → U ⊗ X be a
+Poisson Q-module map; we need to find a unique A-module map θ: U(U, V ) → X such
+that g =
+�
+IdU ⊗θ
+�
+◦ ρV . Let {zsr | s = 1, · · · , m, r ∈ J} be a family of elements of X such
+that for all r ∈ J we have:
+g(vr) =
+m
+�
+s=1
+us ⊗ zsr.
+(30)
+Furthermore, as g: V → U ⊗ X is a Poisson Q-modules map, we can easily prove that
+the following compatibilities hold for all s = 1, · · · , m, i ∈ J and j ∈ I:
+�
+p∈Wj,i
+σp
+j,i zsp =
+m
+�
+t=1
+n
+�
+r=1
+γs
+r,t xrj · zti
+(31)
+�
+p∈Tj,i
+ηp
+j,i zsp =
+m
+�
+t=1
+n
+�
+r=1
+ωs
+r,t xrj · zti
+(32)
+where · denotes the A-module action on X. The universal property of the free module
+yields a unique A-module map θ: U(U, V ) → X such that θ(Ysr) = zsr, for all s =
+1, · · · , m and r ∈ J. Moreover, Ker(θ) contains the A-submodule of U(U, V ) generated
+
+14
+A. L. AGORE AND G. MILITARU
+by the elements listed in (24) and (25). Indeed, as θ: U(U, V ) → X is a morphism of
+A-modules we have:
+θ
+� �
+p∈Wj,i
+σp
+j,i Ysp −
+m
+�
+t=1
+n
+�
+r=1
+γs
+r,t xrj ⋄ Yti
+�
+=
+�
+p∈Wj,i
+σp
+j,i zsp −
+m
+�
+t=1
+n
+�
+r=1
+γs
+r,t xrj · zti
+(31)
+= 0
+θ
+� �
+p∈Tj,i
+ηp
+j,i Ysp −
+m
+�
+t=1
+n
+�
+r=1
+ωs
+r,t xrj ⋄ Yti
+�
+) =
+�
+p∈Tj,i
+ηp
+j,i zsp −
+m
+�
+t=1
+n
+�
+r=1
+ωs
+r,t xrj · zti
+(32)
+= 0
+for all s = 1, · · · , m, i ∈ J and j ∈ I.
+This shows that there exists a unique A-
+module map θ: U(U, V ) → X such that θ(ysr) = zsr, for all s = 1, · · · , m and r ∈ J.
+Furthermore, this implies that for all r ∈ J we have:
+�
+IdU ⊗ θ
+�
+◦ ρV (vr) =
+�
+IdU ⊗ θ
+�� m
+�
+s=1
+us ⊗ ysr
+�
+=
+m
+�
+s=1
+us ⊗ zsr
+(30)
+= g(vr)
+θ is obviously unique with this property and therefore the map ΓV,X is bijective.
+We are left to show that given a finite dimensional Poisson P-module U, assigning
+V �→ U(U, V ) defines a functor U(U, −): QPM → AM. Indeed, let h: V1 → V2 be a
+Poisson Q-modules map. The bijectivity of ΓV1, U(U, V2) applied for the Poisson Q-modules
+map ρV2 ◦h: V1 → U ⊗ U(U, V2), yields a unique A-module map h: U(U, V1) → U(U, V2)
+such that:
+�
+IdU ⊗ h
+�
+◦ ρV1 = ρV2 ◦ h
+By setting U(U, h) to be this unique morphism h, the functor U(U, −) is fully defined.
+Moreover, it can now be easily checked that U(U, −) is indeed a functor and that ΓV, X is
+natural in both variables. Therefore, U(U, −) is the left adjoint of the functor U ⊗−.
+□
+Keeping the notations and the assumptions of Theorem 2.5 we can prove the following:
+Theorem 2.7. Let P and Q be two Poisson algebras such that P is finite dimensional,
+A = P(P, Q) and let V = (V, ·) be a finite dimensional A-module. Then the functor
+− ⊗ V : PPM → QPM has a left adjoint V(V, −): QPM → P PM.
+Proof. Since the proof goes in the same manner as the one of Theorem 2.6, we only
+indicate its main steps. Let {v1, · · · , vm}, m ∈ N∗, be a k-basis of the A-module V and
+denote by γt
+i,j,s ∈ k the structure constants of V , i.e. for all i = 1, · · · , n, j ∈ J and
+s = 1, · · · , m we have:
+xij · vs =
+m
+�
+t=1
+γt
+i,j,s vt
+Let (W, ⊢, ↬) ∈ QPM be a Poisson Q-module and {wr | r ∈ J} its k-basis. For all j ∈ I
+and r ∈ J we can find two finite subsets Sj,r and Tj,r of J such that:
+fj ⊢ wr =
+�
+t∈Sj,r
+σt
+j,r wt,
+fj ↬ wr =
+�
+s∈Tj,r
+ηs
+j,r ws
+
+UNIVERSAL CONSTRUCTIONS FOR POISSON ALGEBRAS
+15
+where σt
+j,r, ηs
+j,r ∈ k, for all j ∈ I, r ∈ J, t ∈ Sj,r and s ∈ Tj,r. Using Remark 1.1 we
+can now define V(V, W) =
+�
+V(V, W), ◮, ↷
+�
+as the free Poisson P-module generated
+by the set {yji | j ∈ J, i = 1, · · · , m} subject to the following relations:
+�
+t∈Sj,r
+σt
+j,r yra =
+n
+�
+i=1
+m
+�
+b=1
+γa
+i,j,b (ei ◮ yrb)
+(33)
+�
+s∈Tj,r
+ηs
+j,r ysa =
+m
+�
+b=1
+n
+�
+i=1
+γa
+i,j,b (ei ↷ yrb)
+(34)
+for all j ∈ I, r ∈ J and a = 1, · · · , m. Now relations (33) and (34) allow us to easily
+prove that the linear map defined for any r ∈ J by:
+ηW : W → V(V, W) ⊗ V,
+ηW (wr) :=
+m
+�
+s=1
+yrs ⊗ vs
+is a morphism of Poisson Q-modules and, analogous to the proof of Theorem 2.6, the
+canonical map
+HomP PM (V(V, W), U) → HomQPM (W, U ⊗ V ),
+θ �→ (θ ⊗ IdV ) ◦ ηW
+is a natural isomorphism for any Poisson Q-module W and any Poisson P-module U.
+The proof is now finished.
+□
+Before giving some examples, it will be useful to observe the following: since the bracket
+on the Lie algebras on P and Q is skew-symmetric we have µs
+i,i = βu
+i,i = 0, µs
+i,j = −µs
+j,i
+and βu
+i,j = −βu
+j,i. Consequently, relations (11) are automatically fulfilled for i = j.
+Examples 2.8. 1. Let P and Q be two Poisson algebras such that P is finite dimensional
+and the associative algebra structures on both P and Q are the trivial ones (i.e. xy := 0,
+for any x, y ∈ P (resp. Q)). Thus P and Q are just Lie algebras viewed as Poisson
+algebras. Then, P(P, Q) is exactly the universal algebra A(P, Q) of the two Lie algebras
+as constructed in [5, Theorem 2.1]. In particular, if the Lie algebras structures on P
+and Q are also the abelian ones, then P(P, Q) ∼= k[Xsi |s = 1, · · · , n, i ∈ I], where
+n = dimk(P) and |I| = dimk(Q).
+In general, P(P, Q) is the quotient of the universal algebra A(P, Q) of the two Lie
+algebras P and Q, through the ideal generated by the relations listed in (10).
+2. Let P := k be the 1-dimensional Poisson algebra, i.e. the constant structures are
+τ 1
+1,1 = 1 and µ1
+1,1 = 0. For any Poisson algebra Q with a k-basis {fi | i ∈ I} and the
+constant structures αu
+i,j, βu
+i,j ∈ k given by (7), the universal algebra P(k, Q) is the algebra
+generated by the commuting variables xi, i ∈ I, subject to the relations for any i, j ∈ I:
+�
+u∈Ui,j
+αu
+i,j xu = xixj,
+�
+u∈Vi,j
+βu
+i,j xu = 0.
+The other way around, let Q := k and P an n-dimensional Poisson algebra with the
+constant structures {τ s
+i,j | i, j, s = 1, · · · , n} and {µs
+i,j | i, j, s = 1, · · · , n} given by (6).
+
+16
+A. L. AGORE AND G. MILITARU
+Then the universal algebra P(P, k) is the algebra generated by the commuting variables
+x1, · · · , xn subject to the relations:
+n
+�
+s,t=1
+τ a
+s,t xsxt = xa,
+n
+�
+s,t=1
+µa
+s,t xsxt = 0
+for all a = 1, · · · , n.
+3. Let k be a field of characteristic ̸= 2, P := k[X]/(X2) viewed as a Poisson algebra
+with the abelian bracket and Q := aff(2, k) the affine 2-dimensional Lie algebra with
+basis {f1, f2} and bracket given by [f1, f2] = f2 viewed as a Poisson algebra with the
+trivial multiplication (xy := 0, for all x, y ∈ Q). Then:
+P
+�
+P, Q
+�
+∼=
+k[X11, X12, X21, X22]/(X2
+11, X12, X11X21, X22)
+∼=
+k[X, Y ]/(X2, XY )
+Indeed, the only non-zero structure constants of P and Q are: τ 1
+1,1 = τ 2
+1,2 = τ 1
+2,1 = 1
+and β2
+1,2 = 1 = −β2
+2,1. A direct computation shows that, among the sixteen compati-
+bilities resulting from the defining relations (10) and (11) of P
+�
+P, Q
+�
+, after eliminating
+the redundant relations the only remaining ones are the following: x2
+11 = 0, x12 = 0,
+2 x11x21 = 0 and x22 = 0. The conclusion now follows.
+3. The universal coacting bialgebra on a finite dimensional Poisson
+algebra. Applications
+Let P be a finite dimensional Poisson algebra having {e1, · · · , en} as a k-basis. The
+description of the commutative algebra P(P) := P(P, P) given by Theorem 2.2 is the
+following: if {τ s
+i,j | i, j, s = 1, · · · , n} and {µs
+i,j | i, j, s = 1, · · · , n} are the structure con-
+stants of P with respect to the associative and Lie structures as given by (6), then P(P)
+is the free commutative algebra generated by {xsi | s, i = 1, · · · , n, } and the relations:
+n
+�
+u=1
+τ u
+i,j xau =
+n
+�
+s,t=1
+τ a
+s,t xsixtj,
+n
+�
+u=1
+µu
+i,j xau =
+n
+�
+s,t=1
+µa
+s,t xsixtj
+(35)
+for all a, i, j = 1, · · · , n. Furthermore, the map
+ηP : P → P ⊗ P(P),
+ηP(ei) :=
+n
+�
+s=1
+es ⊗ xsi,
+for all i = 1, · · · , n
+(36)
+is a Poisson algebra homomorphism. By considering Q := P in the bijection described
+in (13) we obtain:
+Corollary 3.1. Let P be a finite dimensional Poisson algebra. Then for any comutative
+algebra A and any Poisson algebra homomorphism f : P → P ⊗ A, there exists a unique
+algebra homomorphism θ : P(P) → A such that f = (IdP ⊗ θ) ◦ ηP .
+Next we show that the commutative algebra P(P) can be endowed with a bialgebra
+structure such that (P, ηP ) becomes a right Poisson P(P)-comodule algebra.
+
+UNIVERSAL CONSTRUCTIONS FOR POISSON ALGEBRAS
+17
+Proposition 3.2. Let P be a Poisson algebra of dimension n.
+Then there exists a
+unique bialgebra structure on P(P) such that the Poisson algebra homomorphism ηP :
+P → P ⊗ P(P) becomes a right P(P)-comodule structure on P. The comultiplication
+and the counit on P(P) are given by
+∆(xij) =
+n
+�
+s=1
+xis ⊗ xsj
+and
+ε(xij) = δi,j
+(37)
+for all i, j = 1, · · · , n.
+Proof. Consider the Poisson algebra homomorphism (ηP ⊗IdP(P )) ◦ ηP : P → P ⊗P(P)⊗
+P(P). Corollary 3.1 yields a unique algebra homomorphism ∆ : P(P) → P(P) ⊗ P(P)
+such that the following holds:
+(IdP ⊗ ∆) ◦ ηP = (ηP ⊗ IdP(P )) ◦ ηP .
+(38)
+Applying (38) for each ei, i = 1, · · · , n and using (36) we obtain the following:
+n
+�
+t=1
+et ⊗ ∆(xti) = (ηP ⊗ Id)(
+n
+�
+s=1
+es ⊗ xsi) =
+n
+�
+s=1
+(
+n
+�
+t=1
+et ⊗ xts) ⊗ xsi
+=
+n
+�
+t=1
+et ⊗ (
+n
+�
+s=1
+xts ⊗ xsi)
+which comes down to ∆(xti) = �n
+s=1 xts ⊗ xsi, for all t, i = 1, · · · , n. Note that ∆
+is obviously coassociative. In a similar fashion, applying once again Corollary 3.1, we
+obtain a unique algebra homomorphism ε: P(P) → k such that the following holds:
+(IdP ⊗ ε) ◦ ηP = can
+(39)
+where can : P → P ⊗ k is the canonical isomorphism, can(x) = x ⊗ 1, for all x ∈ P.
+Applying (39) for each ei, i = 1, · · · , n, we obtain ε(xij) = δi,j, for all i, j = 1, · · · , n.
+It can be easily checked that ε is a counit for ∆, and therefore P(P) is a bialgebra.
+Furthermore, (38) and (39) imply that the canonical map ηP : P → P ⊗ P(P) defines a
+right P(P)-comodule structure on P.
+□
+The key property of P(P) is the following Poisson algebra version of [5, Theorem 2.11]:
+Theorem 3.3. Let P be a finite dimensional Poisson algebra. Then, (P(P), ηP ) is the
+initial object of the category CoactBialgP of all commutative bialgebras coacting on P
+and we call it the universal coacting bialgebra of P.
+Proof. The statement of the theorem comes down to showing that for any commutative
+bialgebra B and any Poisson algebra homomorphism f : P → P ⊗B which makes P into
+a right B-comodule there exists a unique bialgebra homomorphism θ: P(P) → B such
+that the following diagram is commutative:
+P
+ηP
+�
+f
+�■
+■
+■
+■
+■
+■
+■
+■
+■
+■
+P ⊗ P(P)
+IdP ⊗θ
+�
+P ⊗ B
+(40)
+
+18
+A. L. AGORE AND G. MILITARU
+To start with, using Corollary 3.1, we obtain a unique algebra homomorphism θ: P(P) →
+B such that diagram (40) commutes. The proof will be finished once we show that θ is
+a coalgebra homomorphism as well. This follows by using again Corollary 3.1. Indeed,
+we obtain a unique algebra homomorphism ψ: P(P) → B ⊗ B such that the following
+holds:
+(IdP ⊗ ψ) ◦ ηP =
+�
+IdP ⊗ ∆B ◦ θ
+�
+◦ηP
+(41)
+Obviously the algebra homomorphism ∆B ◦θ: P(P) → B ⊗ B fulfills the above compat-
+ibility. The proof will be finished once we show that (θ ⊗ θ) ◦ ∆: P(P) → B ⊗ B fulfills
+the same compatibility. Indeed, as f : P → P ⊗ B is a right B-comodule structure, we
+have:
+�
+IdP ⊗ (θ ⊗ θ) ◦ ∆
+�
+◦ ηP
+=
+�
+IdP ⊗ θ ⊗ θ
+�
+◦
+�
+IdP ⊗ ∆
+�
+◦ ηP
+(38)
+=
+�
+IdP ⊗ θ ⊗ θ
+�
+◦(ηP ⊗ IdP(P )) ◦ ηP
+=
+�
+(IdP ⊗ θ) ◦ ηP ⊗ θ
+�
+◦ ηP
+(40)
+=
+�
+f ⊗ θ
+�
+◦ ηP
+=
+(f ⊗ IdB) ◦ (IdP ⊗ θ) ◦ ηP
+(40)
+=
+(f ⊗ IdB) ◦ f
+=
+(IdP ⊗ ∆B) ◦ f
+(40)
+=
+(IdP ⊗ ∆B) ◦ (IdP ⊗ θ) ◦ ηP
+=
+(IdP ⊗ ∆B ◦ θ) ◦ ηP
+as desired. Similarly, one can show that εB ◦ θ = ε and the proof is now finished.
+□
+By considering Takeuchi’s commutative Hopf envelope [24] of the bialgebra P(P) we
+obtain, using Theorem 3.3, the following:
+Corollary 3.4. Let P be a finite dimensional Poisson algebra.
+Then the category
+CoactHopfP consisting of all commutative Hopf algebras coacting on P has an initial
+object
+�
+H(P), λP
+�
+and we call it the universal coacting Hopf algebra of P.
+Proof. Indeed, the forgetful functor U : ComHopfk → ComBiAlgk from the category of
+commutative Hopf algebras to the category of commutative bialgebras has a left adjoint
+L: ComBiAlgk → ComHopfk ([24, Theorem 65, (2)]). If we denote by µ: 1ComBiAlgk →
+UL the unit of the adjunction L ⊣ U, then we can easily prove, in the spirit of [5,
+Theorem 2.13], that the pair
+�
+H(P) := L(P(P)), λP := (IdP ⊗µP(P )) ◦ ηP
+�
+is the initial
+object in the category CoactHopfP of all commutative Hopf algebras coacting on P.
+□
+Remark 3.5. The dual versions of Theorem 3.3 and Corollary 3.4 regarding the actions
+of commutative bialgebras (resp. Hopf algebras) on a Poisson algebra also hold. For
+a Poisson algebra P, we can define the category ActBialgP (resp.
+ActHopfP ) of all
+commutative bialgebras (respectively Hopf algebras) which act on P. More precisely, the
+objects of ActBialgP (resp. ActHopfP) are pairs (B, µP) consisting of a commutative
+bialgebra (resp. Hopf algebra) B and a linear map µP : P ⊗ B → P, such that (P, µP )
+
+UNIVERSAL CONSTRUCTIONS FOR POISSON ALGEBRAS
+19
+is a (right) Poisson B-module algebra, i.e. µP is a (right) B-module structure on P as
+well as a Poisson algebra map. Using the same arguments as in [2, Theorem 4.14], we
+can prove that ActBialgP (resp. ActHopfP ) has a final object.
+Next we will present two important applications of the bialgebra P(P).
+These are
+the Poisson algebra version of similar results obtained for Lie/associative algebras in
+[5, 23]. First, recall the well known fact that for any bialgebra H, we have G(Ho) =
+HomAlgk(H, k), the set of all algebra homomorphisms H → k (see ([26, pag. 62])).
+Theorem 3.6. Let P be a finite dimensional Poisson algebra with basis {e1, · · · , en} and
+U
+�
+G
+�
+P(P)o��
+the group of all invertible group-like elements of the finite dual P(P)o.
+Then the map defined for any θ ∈ U
+�
+G
+�
+P(P)o��
+and i = 1, · · · , n by:
+γ : U
+�
+G
+�
+P(P)o��
+→ AutPoiss(P),
+γ(θ)(ei) :=
+n
+�
+s=1
+θ(xsi) es
+(42)
+is an isomorphism of groups.
+Proof. Using Corollary 2.4 for Q := P yields the bijective map
+γ : HomAlgk(P(P), k) → EndPoiss(P),
+γ(θ) =
+�
+IdP ⊗ θ
+�
+◦ηP
+Furthermore, as discussed above we have HomAlgk(P(P), k) = G
+�
+P(P)o�
+and based
+on (36) it follows easily that γ takes the form given in (42).
+We denote by γ the
+restriction of γ to the invertible elements of the two monoids where the monoid structure
+on EndPoiss(P) is given by the usual composition of endomorphisms while G
+�
+P(P)o�
+is
+a monoid with respect to the convolution product, i.e.
+(θ1 ⋆ θ2)(xsj) =
+n
+�
+t=1
+θ1(xst)θ2(xtj)
+(43)
+for all θ1, θ2 ∈ G
+�
+P(P)o�
+and j, s = 1, · · · , n. Therefore, the proof will be finished by
+showing that γ is a monoid isomorphism and this can be shown exactly as in [5, Theorem
+3.1].
+□
+Next, for a given abelian group G, we describe all G-gradings on a Poisson algebra P.
+Proposition 3.7. Let G be an abelian group and P a finite dimensional Poisson algebra.
+There exists a bijection between the set of all G-gradings on P and the set of all bial-
+gebra homomorphisms P(P) → k[G] given such that the G-grading on P = ⊕σ∈G P (θ)
+σ
+associated to a bialgebra map θ : P(P) → k[G] can be described as follows:
+P (θ)
+σ
+:= {x ∈ P |
+�
+IdP ⊗ θ
+�
+◦ ηP (x) = x ⊗ σ}
+(44)
+for all σ ∈ G.
+Proof. Theorem 3.3 applied for the commutative bialgebra B := k[G] yields a bijection
+between the set of all bialgebra homomorphisms P(P) → k[G] and the set of all Poisson
+algebra homomorphisms f : P → P ⊗ k[G] which make P into a right k[G]-comodule.
+The proof is now finished since we have shown in Example 1.2 that the latter set is in
+bijective correspondence with the set of all G-gradings on the Poisson algebra P.
+□
+
+20
+A. L. AGORE AND G. MILITARU
+Our next aim is to classify all G-gradings on a Poisson algebra P.
+To this end, we
+introduce the following:
+Definition 3.8. Let G be an abelian group and P a finite dimensional Poisson algebra.
+Two homomorphisms of bialgebras θ1, θ2 : P(P) → k[G] are called conjugate and denote
+this by θ1 ≈ θ2, if there exists g ∈ U
+�
+G
+�
+P(P)o��
+an invertible group-like element of the
+finite dual P(P)o such that θ2 = g⋆θ1⋆g−1, in the convolution algebra Hom
+�
+P(P), k[G]
+�
+.
+Throughout, HomBiAlg
+�
+P(P), k[G]
+�
+/ ≈ will denote the quotient set of the set of all
+bialgebra homomorphisms P(P) → k[G] by the above equivalence relation and let ˆθ
+denote the equivalence class of θ ∈ HomBiAlg
+�
+P(P), k[G]
+�
+. The next theorem classifies
+all G-gradings on a Poisson algebra P.
+Theorem 3.9. Let G be an abelian group, P a finite dimensional Poisson algebra and
+G-gradings(P) the set of isomorphism classes of all G-gradings on P. Then the map
+HomBiAlg
+�
+P(P), k[G]
+�
+/ ≈ �→ G−gradings(P),
+ˆθ �→ P (θ) := ⊕σ∈G P (θ)
+σ
+where P (θ)
+σ
+= {x ∈ P |
+�
+IdP ⊗ θ
+�
+◦ ηP (x) = x ⊗ σ}, for all σ ∈ G, is bijective.
+Proof. Since the associative and Lie/Leibniz algebra counterparts of this result have been
+proved in detail in [23, Theorem 3.4] and [5, Theorem 3.5], respectively, we will be brief.
+First, note that by Proposition 3.7, for any G-grading P = ⊕σ∈G Pσ there exists a unique
+bialgebra homomorphism θ : P(P) → k[G] such that Pσ = P (θ)
+σ , for all σ ∈ G. The
+proof will be finished once we show that any two G-gradings on P, say P (θ1) and P (θ2),
+associated to two bialgebra homomorphisms θ1, θ2 : P(P) → k[G], are isomorphic if and
+only if θ1 ≈ θ2. Indeed, recall from Example 1.2 that defining a G-grading on P is in one-
+to-one correspondence to defining a right k[G]-comodule structure ρ: P → P ⊗k[G] on P
+which is also a Poisson algebra homomorphism. Now two G-gradings P (θ1) and P (θ2) are
+isomorphic if and only if (P, ρ(θ1)) and (P, ρ(θ2)) are isomorphic both as algebras and as
+right k[G]-comodules; this comes down to the existence of an automorphism w: P → P
+of the Poisson algebra P such that ρ(θ2) ◦ w =
+�
+w ⊗ Idk[G]
+�
+◦ρ(θ1). By Theorem 3.6, for
+any Poisson algebra automorphism w : P → P there exists a unique invertible group-
+like element of the finite dual g ∈ U
+�
+G
+�
+P(P)o��
+such that w = wg is given for any
+i = 1, · · · , n by
+wg(ei) =
+n
+�
+s=1
+g(xsi) es
+(45)
+where {e1, · · · , en} is a basis in P. A straightforward computation shows that the Poisson
+algebra automorphism wg : P → P is also a right k[G]-comodule map if and only if the
+following holds:
+n
+�
+s=1
+g(xas)θ1(xsi) =
+n
+�
+s=1
+θ2(xas)g(xsi)
+(46)
+Having in mind that {xai}a,i=1,··· ,n is a system of generators of P(P)) we can easily
+conclude that (46) reduces to g ⋆ θ1 = θ2 ⋆ g. This finishes the proof as g: P(P) → k
+is an invertible element in the convolution algebra Hom
+�
+P(P), k[G]
+�
+which shows that
+θ1 ≈ θ2.
+□
+
+UNIVERSAL CONSTRUCTIONS FOR POISSON ALGEBRAS
+21
+We will give now an explicit example which describes the initial object in the category
+of all commutative bialgebras that coacts on a certain 3-dimensional Poisson algebra.
+Example 3.10. Let P be the 3-dimensional Poisson algebra with k-basis {e1, e2, e3}
+and Poisson algebra structure given by e2
+1 := e2, [e1, e3] := e3 (undefined multiplications
+and brackets are all zero). Then, there exists an isomorphism of bialgebras
+P(P) ∼= k[X, Y, Z, T]/(T − XT)
+where the latter has the following bialgebra structure:
+∆( �
+X) = �
+X ⊗ �
+X,
+ε( �
+X) = 1
+∆(�Y ) = �Y ⊗ �
+X + �
+X2 ⊗ �Y ,
+ε(�Y ) = 0
+∆( �Z) = �Z ⊗ �
+X + �T ⊗ �Z,
+ε( �Z) = 0
+∆( �T) = �T ⊗ �T,
+ε( �T) = 1
+The canonical coaction ηP : P → P ⊗ k[X, Y, Z, T]/(T − XT) of this bialgebra on P is
+given by:
+ηP (e1) = e1 ⊗ �
+X + e2 ⊗ �Y + e3 ⊗ �Z
+ηP (e2) = e2 ⊗ �
+X2,
+ηP(e3) = e3 ⊗ �T.
+Indeed, note first that the only non-zero structure constants of P are: τ 2
+1,1 = 1 and
+µ3
+1,3 = 1 = −µ3
+3,1. Now, a careful analysis of the 54 defining relations of P(P) arising
+from (35), leads to the conclusion that after eliminating the redundant ones, we are left
+with the following:
+x12 = 0,
+x13 = 0,
+x23 = 0,
+x32 = 0,
+x22 = x2
+11,
+x33 = x11x33.
+The conclusion now follows by denoting �
+X = x11, �Y = x21, �Z = x31 and �T = x33.
+References
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+Max Planck Institut f¨ur Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
+Simion Stoilow Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700
+Bucharest, Romania
+Email address: [email protected]
+Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei
+14, RO-010014 Bucharest 1, Romania
+Simion Stoilow Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700
+Bucharest, Romania
+Email address: [email protected] and [email protected]
+

I9E4T4oBgHgl3EQfhQ2p/content/tmp_files/2301.05124v1.pdf.txt

@@ -0,0 +1,1904 @@
+Poses of People in Art: A Data Set for Human Pose Estimation in
+Digital Art History
+STEFANIE SCHNEIDER and RICARDA VOLLMER, Ludwig Maximilian University of Munich, Germany
+Throughout the history of art, the pose—as the holistic abstraction of the human body’s expression—has proven to be a
+constant in numerous studies. However, due to the enormous amount of data that so far had to be processed by hand, its crucial
+role to the formulaic recapitulation of art-historical motifs since antiquity could only be highlighted selectively. This is true
+even for the now automated estimation of human poses, as domain-specific, sufficiently large data sets required for training
+computational models are either not publicly available or not indexed at a fine enough granularity. With the Poses of People in
+Art data set, we introduce the first openly licensed data set for estimating human poses in art and validating human pose
+estimators. It consists of 2,454 images from 22 art-historical depiction styles, including those that have increasingly turned
+away from lifelike representations of the body since the 19th century. A total of 10,749 human figures are precisely enclosed by
+rectangular bounding boxes, with a maximum of four per image labeled by up to 17 keypoints; among these are mainly joints
+such as elbows and knees. For machine learning purposes, the data set is divided into three subsets—training, validation, and
+testing—, that follow the established JSON-based Microsoft Common Objects in Context (COCO) format, respectively. Each
+image annotation, in addition to mandatory fields, provides metadata from the art-historical online encyclopedia WikiArt.
+With this paper, we elaborate on the acquisition and constitution of the data set, address various application scenarios, and
+discuss prospects for a digitally supported art history. We show that the data set enables the comprehensive investigation of
+body phenomena in art, whether at the level of individual figures, which can thus be captured in their subtleties, or entire
+figure constellations, whose position, distance, or proximity to one another is considered.
+CCS Concepts: • Information systems → Recommender systems; Image search; • Computing methodologies →
+Object detection; Interest point and salient region detections; • Applied computing → Fine arts.
+Additional Key Words and Phrases: data set, human detection, human pose estimation, digital art history
+1
+INTRODUCTION
+The abstracted human body, into which measurements, proportions, and movements are inscribed, has played a
+crucial role throughout the history of art. This particularly applies to the drawing apprenticeship [61], whose
+best-known example is Leonardo da Vinci’s Vitruvian Man. As early as the 17th century, artists began to structure
+the human pose1 into a ‘language’ of non-verbal communication [43], pursued with scientific meticulousness
+into the 18th century, e.g., by the Physiognomist Johann Caspar Lavater [23]. Attempts to establish a kind of pose
+vocabulary, however, have been made primarily in relation to hand gestures [1, 8], with references to antiquity
+evident in most efforts [6]. It was the Finnish art historian Johan Jakob Tikkanen who, in the 19th century, then
+sought to motivate a differentiated terminology of leg positions [77], drawing on perspectives from the natural
+sciences, such as Darwin’s essays on the expression of humans and animals [19] as well as botanical classification
+systems [37]. In contrast, the studies of the art historian and cultural theorist Aby Warburg at the beginning of
+the 20th century should not be understood as standardized [57]: through his concept of ‘Pathosformeln,’ Warburg
+rather loosely examined body phenomena recurring since antiquity [82, 83].
+This high selectivity of art-historical research—especially when compared to other body-oriented disciplines
+such as theater and dance studies [45, 62]—can be attributed to various reasons. We perceive two factors as pivotal:
+(i) the enormous amount of data that for a comprehensive analysis so far had to be processed by hand, and (ii) the
+lack of an approach that holistically and systematically assesses human pose through relevant keypoints, e.g.,
+1For reasons of simplicity, we hereinafter do not distinguish between the terms ‘posture’ and ‘pose.’ Instead, we use the term ‘pose’ for any
+kind of bodily expression.
+Authors’ address: Stefanie Schneider, [email protected]; Ricarda Vollmer, [email protected], Ludwig
+Maximilian University of Munich, Geschwister-Scholl-Platz 1, Munich, Germany.
+arXiv:2301.05124v1  [cs.CV]  12 Jan 2023
+
+2
+•
+S. Schneider and R. Vollmer
+Fig. 1. We differentiate between two annotation modes: bounding box and keypoint annotation. First, as shown on the left
+in Andrea del Sarto’s Pietà with Saints (1523–1524), human figures are marked with bounding boxes enclosing them. For a
+maximum of four per image, up to 17 pose-relevant keypoints are then assigned, which are indicated with green circles in
+the detail view on the right.
+wrists or knees. With the ongoing digitization and online publication of historical objects, researchers could now
+potentially draw on increasingly large collections of images to examine dominant pose types or time-dependent
+body phenomena. To date, however, few approaches to automatically estimate human poses in art-historical
+imagery have emerged [33–35, 48, 49], possibly due to the lack of domain-specific, sufficiently large data sets
+required for training computational models, e.g., Convolutional Neural Networks (CNNs). Existing data sets fall
+broadly into two categories. Either they do index keypoints but are not publicly available and are dedicated to a
+comparatively narrow subset of art-historical representation practices [34, 48]. Or they are freely accessible to
+the public but enclose human figures only by rectangular bounding boxes; their pose is then broadly categorized
+without specifically delineating keypoints [64].
+Our contributions are three-fold. (i) With Poses of People in Art, hereinafter abbreviated to PoPArt, we introduce
+the first publicly available and openly licensed data set for estimating human poses in art. It is composed of
+10,749 bounding box and 56,154 keypoint annotations from 22 art-historical depiction styles, including those
+that have emerged since the 19th century and have increasingly turned away from lifelike representations of
+the body; Fig. 1 illustrates both annotation modes. (ii) We demonstrate that PoPArt enables the quantitatively
+systematized exploration of human pose in visual art by capturing the body holistically and across different
+stylistic periods. Pose may thus emerge as wholly elemental to the formulaic recapitulation of significant topoi
+and motifs through computational assistance. (iii) As a by-product of PoPArt’s domain-specific curation, the sole
+detection of figures in art-historical collections is decisively improved. In contrast to the similarly constituted
+
+Poses of People in Art
+•
+3
+People-Art data set [84], which also exclusively labels human figures, PoPArt contains fewer training, validation,
+and testing images. It, however, features nearly three times as many positive training samples with at least one
+figure instance annotation.
+The remainder of this paper is structured as follows. In Section 2, we first review art-historically relevant data
+sets that can be leveraged for image classification and object detection tasks. Section 3 then elaborates on the
+acquisition and constitution of the PoPArt data set. In this context, we also clarify the annotation guidelines we
+adapted to the domain. In the course of Section 4, we address various application scenarios and discuss prospects
+for a digitally supported art history. Lastly, Section 5 concludes the paper and outlines areas for potential future
+research. The data set is available as a version-controlled repository on Zenodo.2
+2
+RELATED WORK
+With the advent of increasingly powerful deep-learning architectures in recent years, the range of domains
+utilizing computational models has expanded decisively. In the field of Computer Vision, e.g., not only real-
+world imagery is dealt with anymore, but also figurative representations of imagined phenomena, which are
+prevalent in art, and across various phases of art history. However, due to those collections’ highly original
+visuals, domain-specific, sufficiently large data sets are still required for training and fine-tuning models.
+Prior to the creation of the PoPArt data set, we conducted an extensive study, aggregated in Table 1, reviewing
+existing art-historical data sets that can be leveraged for image classification and object recognition tasks. Neither
+did we consider data sets featuring solely contemporary or born-digital art [86], nor cultural institutions that,
+while offering relevant data on their websites, do not explicitly make them available in downloadable form, but
+require prior harvesting.3 We also excluded data sets that are exclusively applicable to other research areas like
+aesthetic quality assessment [2], sentiment analysis [54, 88], or correspondence matching [35, 70]. While formal
+attributes at the image-level are contained in a large number of data sets, enabling the classification of artists,
+materials, or creation dates, among others [5, 41, 46, 50, 52, 55, 75, 76, 85, 90], content-based tags are less frequent.
+This is due to the fact that labels referring to the image phenomena actually shown must be determined by
+manual annotation, driven either by crowdsourcing approaches [4] or singular institutional efforts [16, 29, 59].
+The latter rely on the iconographic classification system Iconclass, which is conceived for the Western motifs
+of the visual arts [78]. As a result of the already time-consuming labeling process at image-level, few data sets
+feature object-level annotations [3, 15, 29, 34, 48, 64, 84, 89]. When provided, they are usually marked with
+bounding boxes, so that object instances are enclosed with rectangles and thus precisely located in the image. To
+our work here of particular importance is the People-Art data set [84], in which human figures shown in nearly
+1,500 images are labeled with bounding boxes. Unlike the ten times larger DEArt data set [64], which identifies
+figures in collections only from the 12th to 18th centuries, People-Art indicates depiction styles that encompass
+Impressionist movements as well as Surrealist ones with rather artificial forms of body representation.
+For the decoding of human poses, the rectangular framing of the entire body is not sufficient: individual limbs
+cannot be identified and differentiated any more than joints, such as elbows and wrists. To obtain more accurate
+information about the position of articulation points, three annotation practices have been used. Reshetnikov
+et al. [64] roughly classify poses into 12 categories, e.g., by labeling human figures as sitting or kneeling. Carneiro
+et al. [15], on the other hand, place additional bounding boxes around the torso and head to approximate the
+specifics of the human body. Only Impett and Süsstrunk [34] and Madhu et al. [48], however, apply fine-grained
+labels to faithfully represent bodily specifics by assigning keypoints on areas relevant to the figure’s pose, e.g.,
+the hips, knees, or ears. In doing so, they adhere to labeling techniques common for real-world human pose
+2https://doi.org/10.5281/zenodo.7516230.
+3For institutions from the GLAM (Galleries, Libraries, Archives, and Museums) sector that have published open access data, see the following
+survey: https://docs.google.com/spreadsheets/d/1WPS-KJptUJ-o8SXtg00llcxq0IKJu8eO6Ege_GrLaNc/edit.
+
+4
+•
+S. Schneider and R. Vollmer
+Table 1. Art-historically relevant data sets for image classification and object detection tasks are compared. Grey check
+marks specify information that is not directly stored in the respective data set, but has to be accessed via the referenced
+content providers.
+Name
+Author(s)
+Year
+Annotation
+Levels
+Availability
+Formal
+Content
+Image
+Object
+Public
+Privat
+Medieval Manuscripts [89]
+Yarlagadda et al.
+2010
+✓
+✓
+✓
+✓1
+✓
+WikiArt (f.k.a. WikiPaintings) [85]
+Unknown
+2010
+✓
+✓
+✓
+PrintART [15]
+Carneiro et al.
+2012
+✓
+✓
+✓1
+✓
+Paintings [18]
+Crowley and Zisserman
+2014
+✓
+✓
+✓
+✓
+Picasso [28]
+Ginosar et al.
+2014
+✓
+✓
+✓
+✓
+Painting-91 [41]
+Khan et al.
+2014
+✓
+✓
+✓
+Rijksmuseum Challenge [52]
+Mensink and van Gemert
+2014
+✓
+✓
+✓
+Pandora [24]
+Florea et al.
+2016
+✓
+✓
+✓
+Warburg’s Bilderatlas [34]
+Impett and Süsstrunk
+2016
+✓
+✓
+✓
+✓1,2
+✓
+Painter by Numbers [55]
+Nichol
+2016
+✓
+✓
+✓
+Visual Link [69]
+Seguin et al.
+2016
+✓
+✓
+✓
+People-Art [84]
+Westlake et al.
+2016
+✓
+✓
+✓
+✓1
+✓
+Art500k [50]
+Mao et al.
+2017
+✓
+✓
+✓
+BibleVSA [3]
+Baraldi et al.
+2018
+✓
+✓
+✓
+✓1
+✓
+ARTigo [4]
+Becker et al.
+2018
+✓
+✓
+✓
+✓
+SemArt [26]
+Garcia and Vogiatzis
+2018
+✓
+✓
+✓
+✓
+IconArt [29]
+Gonthier et al.
+2018
+✓
+✓
+✓
+✓1
+✓
+OmniArt [76]
+Strezoski and Worring
+2018
+✓
+✓
+✓
+MultitaskPainting100k [5]
+Bianco et al.
+2019
+✓
+✓
+✓
+Ancient Chinese Art [71]
+Sheng and Moens
+2019
+✓
+✓
+✓
+✓
+Ancient Egyptian Art [71]
+Sheng and Moens
+2019
+✓
+✓
+✓
+✓
+Artpedia [75]
+Stefanini et al.
+2019
+✓
+✓
+✓
+✓
+Iconclass Caption [16]
+Cetinic
+2021
+✓
+✓
+✓
+AQUA [27]
+Garcia et al.
+2020
+✓
+✓
+✓
+✓
+ClassArch [48]
+Madhu et al.
+2020
+✓
+✓
+✓
+✓1,2
+✓
+Iconclass AI Test Set [59]
+Posthumus
+2020
+✓
+✓
+✓
+Saints [66]
+Schneider et al.
+2020
+✓
+✓
+✓
+✓
+ArtDL [53]
+Milani and Fraternali
+2021
+✓
+✓
+✓
+The Met [90]
+Ypsilantis et al.
+2021
+✓
+✓
+✓
+ArtBench-10 [46]
+Liao et al.
+2022
+✓
+✓
+✓
+DEArt [64]
+Reshetnikov et al.
+2022
+✓
+✓
+✓
+✓1
+✓
+PoPArt
+Schneider and Vollmer
+2023
+✓
+✓
+✓
+✓1,2
+✓
+1Object-level annotations include bounding boxes.
+2Object-level annotations include keypoints.
+estimation. The Microsoft Common Objects in Context (COCO) format guidelines, for instance, require that
+17 keypoints be stored with their 𝑥𝑦-coordinates.4 Both data sets suffer from two issues: they are (i) not made
+publicly available for further reuse, and (ii) devoted to only a comparatively narrow subset of art-historical modes
+of depicting human figures; Impett and Süsstrunk [34] extracted panels from Warburg’s Bilderatlas Mnemosyne,
+whereas Madhu et al. [48] focused on ancient Greek vase paintings. With PoPArt, we address this desideratum
+and introduce the first publicly available data set for human pose estimation in art-historical figures, covering
+4https://cocodataset.org/#format-data.
+
+Poses of People in Art
+•
+5
+Table 2. Figure detection results are reported for the People-Art test set [84]. For training and validation, People-Art is used
+as well. In contrast to previous benchmarks by Kadish et al. [36] and Gonthier et al. [30], we include difficult-to-annotate
+figures. The best performing approach is indicated in bold.
+Model
+Backbone
+LR
+AP
+AP50
+AP75
+AP𝑆
+AP𝑀
+AP𝐿
+AR
+TOOD [22]
+ResNet-50-FPN
+2e − 4
+0.461
+0.750
+0.490
+0.197
+0.296
+0.493
+0.635
+PVT [81]
+PVTv2-B2
+1e − 5
+0.465
+0.760
+0.484
+0.060
+0.263
+0.505
+0.601
+Cascade R-CNN [11]
+ResNet-50-FPN
+2e − 4
+0.444
+0.758
+0.468
+0.147
+0.297
+0.476
+0.593
+SABL Cascade R-CNN [80]
+ResNet-50-FPN
+2e − 4
+0.443
+0.741
+0.458
+0.139
+0.286
+0.476
+0.593
+Faster R-CNN [63]
+ResNet-50-FPN
+2e − 4
+0.423
+0.749
+0.421
+0.115
+0.298
+0.450
+0.568
+SABL Faster R-CNN [80]
+ResNet-50-FPN
+2e − 4
+0.441
+0.752
+0.466
+0.123
+0.284
+0.475
+0.596
+PISA Faster R-CNN [13]
+ResNet-50-FPN
+2e − 4
+0.434
+0.753
+0.451
+0.137
+0.290
+0.463
+0.568
+Libra Faster R-CNN [56]
+ResNet-50-FPN
+2e − 4
+0.417
+0.747
+0.416
+0.068
+0.290
+0.445
+0.569
+impressionistic to neo-figurative and realistic depiction styles. Since our data set follows the Microsoft COCO
+format [47], in addition to bounding boxes, up to 17 keypoints are stored per figure. Five keypoints are provided
+for the head, indicating the nose, eyes, and ears; six for the upper body, indicating wrists, elbows, and shoulders;
+and another six for the lower body, indicating ankles, knees, and hips.
+3
+DATA SET
+This section elaborates on the acquisition and constitution of the PoPArt data set. First, we outline the image
+collection (Section 3.1) and annotation procedures (Section 3.2). We then provide an in-depth statistical analysis
+of the data set (Section 3.3) and present its underlying data format (Section 3.4).
+3.1
+Image Collection
+Like many authors before, e.g., Westlake et al. [84] and Mao et al. [50], we exploit the art-historical online
+encyclopedia WikiArt [85] as content provider. This decision is attributable to several factors: (i) reproductions
+provided in WikiArt are mostly in the public domain and can thus be redistributed under free licenses; (ii) not only
+does WikiArt embrace the widely received canon of Western art history, but does also include Eastern movements,
+such as the early 20th-century Japanese Shin-hanga, albeit to a much lesser extent; (iii) because WikiArt stores
+the depiction style of each object, fine-grained evaluations are facilitated, even if such classifications are to be
+understood as loose, arbitrary, or possibly biased constructs [12, 20].
+To further ensure that PoPArt is representative of both the projective and denotational styles prevalent in
+the domain [87], a semi-automatic data collection procedure was preferred. In a preliminary step, we extracted
+images from WikiArt that have a high probability of depicting human figures, i.e., images on which at least one
+figure can be automatically detected with a probability of 𝑝 = 0.5. To this end, we benchmarked the suitability
+of models commonly used for object detection and applied the best-performing one. The selection ranges from
+multi-stage Region-based Convolutional Neural Networks (R-CNNs) [11, 13, 56, 63, 80] and Transformer-based
+architectures [81] to task-aligned one-stage methods [22]. For evaluation, we use the metrics and tools provided
+by the COCO API.5 All models were first pre-trained on the Microsoft COCO 2017 data set6 for 12 epochs.
+As optimization algorithms, we employed Stochastic Gradient Descent (SGD) for ResNet-50 and Adam [42]
+for Transformer backbones; momentum and weight decay were set to 0.9 and 1e − 4, respectively. The initial
+5https://github.com/cocodataset/cocoapi.
+6https://www.kaggle.com/datasets/awsaf49/coco-2017-dataset.
+
+6
+•
+S. Schneider and R. Vollmer
+(a) Abstract Expressionism
+(b) Art Nouveau
+(c) Baroque
+(d) Contemporary Realism
+(e) Cubism
+(f) Early Renaissance
+(g) Expressionism
+(h) Fauvism
+(i) High Renaissance
+(j) Impressionism
+(k) Mannerism
+(l) Naive Art
+(m) New Realism
+(n) Northern Renaissance
+(o) Pointillism
+(p) Pop Art
+(q) Post Impressionism
+(r) Realism
+(s) Rococo
+(t) Romanticism
+(u) Symbolism
+(v) Ukiyo-e
+Fig. 2. The PoPArt data set contains 22 depiction styles, ranging from impressionistic to neo-figurative and realistic variants.
+For each style, an exemplary image is shown. All images originate from the art-historical online encyclopedia WikiArt [85]
+and are in the public domain.
+
+Poses of People in Art
+•
+7
+(a) Data set view
+(b) Annotation view
+Fig. 3. The web-based open-source tool COCO Annotator [7] provides a light-weight interface that can be used collaboratively
+for annotating bounding boxes and keypoints.
+learning rate decays at the 8th and 11th epoch with 2e − 2 set for ResNet-50-backed and 1e − 4 for Transformer-
+backed architectures. Models were then fine-tuned, with their classification head re-initialized, for another 12
+epochs on People-Art [84]. The learning rate is decreased to 2e − 4 in case of ResNet-50 and 1e − 5 in case of
+Transformer backbones. During training, we adopted the following data augmentation techniques from the
+Albumentations library [9] to increase the models’ robustness: (i) either RandomBrightnessContrast or CLAHE
+is applied with a probability of 𝑝 = 0.2; (ii) either RGBShift or HueSaturationValue is applied with 𝑝 = 0.1;
+(iii) JpegCompression is applied with 𝑝 = 0.2; (iv) ChannelShuffle is applied with 𝑝 = 0.1; and (v) either Blur
+or MedianBlur is applied with 𝑝 = 0.1. Images are reduced to a maximum scale of 1, 333 × 800 pixels without
+changing the aspect ratio. In contrast to previous studies by Kadish et al. [36] and Gonthier et al. [30], we include
+difficult-to-annotate figures. As evident by the benchmark results shown in Table 2, state-of-the-art models such
+as TOOD [22] and PVT [81] outperform multistage R-CNNs to a nearly similar extent in Average Precision (AP)
+between 1.7 and 4.8 %. At a more restrictive Intersection over Union (IoU) threshold of 0.75, the difference
+increases further, rising to between 1.6 and 7.4 %. This effect also is noticeable with Average Recall (AR), which is
+0.5 to 6.8 % higher. Since TOOD surpasses PVT in AR by 3.4 %, with AP being almost equal, we assume that it is
+generally suited best to the stylistic peculiarities of the art-historical domain.
+After pre-filtering the data for images with human figures, we identified the 22 most frequently observed
+depiction styles, covering impressionistic, neo-figurative, and realistic movements from the 14th to the 20th
+century. The integration of data from the 19th and 20th centuries is of particular importance here, as formal
+conventions of bodily phenomena were successively disrupted at the end of the 19th century [10]. We deemed
+22 styles to be adequate to both capture the wide diversity of art-historical image specifics in a time-efficient
+manner, and to later sufficiently assess the validity of computational models for bounding box and keypoint
+estimation depending on the depiction style. A maximum of 125 images per style were then selected for image
+annotation, taking into account the sampling distribution. Exact-duplicate and near-duplicate reproductions were
+removed. For each style, an example image is shown in Fig. 2.
+3.2
+Image Annotation
+The practice of image annotation is characterized by two modes of determinations: whether a human figure can
+be recognized in an image (bounding box annotation) and how his or her pose can be abstracted in it (keypoint
+annotation). Following Everingham et al. [21], we designed the annotation procedure to be as (i) exhaustive,
+(ii) consistent, and (iii) accurate as possible, without omitting art-historical depiction specifics. With COCO
+
+COCO ANNOTATOR
+ DATASETS
+ CATEGORIES
+ TASKS
+ SSCHNEIDER
+POPART
+（2454）IMAGES
+MEMBERS
+STATISTICS
+EXPORTS
+ POPART
+ DATA
+ Identfier
+1148
+1149
+ piero-della-francesca_st-sigismund-and-sigismondo-pandolf...
+O edward-hopper_new-york-restaurant.jpg
+ kuzma-petrov-vodkin_costume-design-for-the-tragedy-of-pu..
+O marc-chagal_adam-and-eve-with-the-forbidden-fruit-1960.j..
+ 2 annotations 
+ 34 keypoints
+7 annotations（12 keypoints
+1 annotation （17 keypoints
+ 2 annotations 
+33 keypoints
+1154
+ oswaldo-guayasamin_from-la-edad-de-la-ternura-series-1.jpg 
+O titian_virgin-and-child.jpg
+O paolo-veronese_venus-and-adonis.jpg
+1 annotation
+(8 keypoints 
+2 annotations（26 keypoints
+3 annotations
+46 keypoints 
+1 annotation
+)（13 keypoints 
+(159
+1160
+(1161
+O theophrastos-triantafyllidis_friends.jpg
+O zinaida-serebriakova_portrait-of-aleksandr-serebriakov-stud... :
+O vladimir-borovikovsky_portrait-of-a-and-v-gagarin-1802.jpg
+O maurice-de-vlaminck_portrait-of-a-woman.jpg
+5 annotations（15 keypoints
+1 annotation（9 keypoints
+2 annotations（26 keypoints
+1 annotation（7 keypoints
+00..121314
+(15 16 17)
+480COCO ANNOTATOR
+DATASETSPOPARTCATEGORIES
+U SSCHNEIDER
+。
+ tintoret_the-birth-fjohn-the-baptist jpg
+QPERSON(11)
+tintoretto_the-birth-of-john-the-baptist.jpg
+2730x1821
+1(ID:9)
+2(ID: 10)
+7(ID: 15
+Q9(ID:17)
+Q10(ID: 18)
+Q11(ID:19)
+O NOSE
+ LEFT_EYE
+ RIGHT_EYE
+ LEFT_EAR
+ RIGHT_EAR
+ Mannerism Late Renaissance8
+•
+S. Schneider and R. Vollmer
+Challenges
+Variations of the size
+of human figures
+Large crowds
+Small figures
+Figures hard to separate
+from each other
+Figures difficult to
+recognize as such
+Image-extrinsic factors
+Image-intrinsic factors
+Figures in the
+background
+Relation of human
+figures to each other
+Referencing
+Shadows
+Reflections
+In water
+In the mirror
+On other surfaces
+(such as helmets)
+Not referencing
+Overlaps
+Intersections
+Symmetrically
+arranged figures
+Deviations from the
+‘ideal’ human body
+Stylistic variance
+Lack of differentiation
+of the face
+Lack of differentiation
+of the body shape
+Veiled body
+Non-human bodies
+and body parts
+Human-like animals
+Mythological figures
+Biblical figures
+Non-living bodies
+and body parts
+Fabricated bodies
+Dolls
+Masks
+Crafts
+Sculptures
+Skeletons
+Severed heads
+Severed limbs
+Image-extrinsic factors
+Image-intrinsic factors
+Positioning of the
+human body
+Back views
+Profile views
+Distortions
+Twists and turns
+Fig. 4. Four aspects pose challenges to the annotation of art-historical imagery: (i) the size of human figures, (ii) their relation
+to each other, (iii) deviations from the ‘ideal’ human body, and (iv) the positioning of the body.
+Annotator [7], we used a web-based open-source tool for bounding box and keypoint annotation that we minimally
+adapted to our needs (Fig. 3).
+3.2.1
+Exhaustiveness. We set the following guidelines to guarantee exhaustive annotation. (i) All human-
+appearing figures are enclosed by bounding boxes; the distance to the outline of the human figure is to be
+kept as small as possible. Only the visible area of the figure is labeled and not the estimated total extent of it.
+Larger numbers of people, whose individual figures can no longer be sufficiently differentiated, are labeled as
+‘crowd.’ In contrast to the Microsoft COCO [47] and PASCAL Visual Object Classes (VOC) data sets [21], we
+
+Poses of People in Art
+•
+9
+Variations of the size
+of human figures
+(a)
+(b)
+(c)
+(d)
+Relation of human
+figures to each other
+(e)
+(f)
+(g)
+(h)
+Deviation from the
+‘ideal’ human body
+(i)
+(j)
+(k)
+(l)
+Positioning of
+the human body
+(m)
+(n)
+(o)
+(p)
+Fig. 5. Sample images of the PoPArt data set illustrate the four aspects that pose challenges to the annotation of art-historical
+imagery: (i) the size of human figures, (ii) their relation to each other, (iii) deviations from the ‘ideal’ human body, and (iv) the
+positioning of the body. All images are in the public domain.
+do not indicate truncated or difficult-to-annotate figures as such. (ii) Up to four human figures per image are
+fine-granularly labeled with keypoints, selecting those whose limbs can be captured best. We do not consider it
+beneficial to label all figures with keypoints, as this would favor styles that feature an above-average number of
+figures—and thus would introduce data bias. Keypoints are recorded in a ‘person-centric’ way, i.e., left points
+refer to the figure’s left extremities. Since in many cases keypoints are not clearly visible or are occluded, we
+establish three rules. (a) If an occluded body part can be approximated by another, it is denoted by a keypoint; e.g.,
+an elbow obscured by a pillar is annotated if the hand and shoulder of the respective body half are visible. (b) Due
+to the low variance of the body parts, eyes and ears are labeled in profile views on the non-visible side of the face
+as well. (c) If several joints are not visible and cannot be approximated, the corresponding keypoints are not set.
+3.2.2
+Consistency. To ensure consistency in the annotation, a fixed team of annotators was employed at the
+Ludwig Maximilian University of Munich throughout the entire period. Annotation guidelines were discussed
+with the annotators prior to annotation and iteratively modified during the annotation procedure, e.g., when
+unusual figure constellations occurred more frequently. In the course of the process, recurring challenges arose
+for both modes, bounding box and keypoint annotation; Fig. 4 visualizes them in taxonomic form. We identify
+four major challenges: (i) those resulting from variations of the size of human figures, (ii) those emerging from
+
+林
+2
+里10
+•
+S. Schneider and R. Vollmer
+the relation of human figures to each other, (iii) those attributable to deviations from the ‘ideal’ human body, and
+(iv) those originating from the body’s positioning in the image space.
+Variations of the size of human figures. Large crowds and figures in the background complicate the annotation.
+Both cases are dominated by very small figures (Fig. 5a; Fig. 5b), figures that are difficult to separate from each
+other (Fig. 5c), or that are difficult to recognize as human (Fig. 5d). The latter is due not only to factors intrinsic
+to the object, i.e., the analog original, but also to image-extrinsic factors, i.e., the original’s digital reproduction.
+In particular, compression artifacts or low-quality and out-of-date resolutions hamper the process.
+Relation of human figures to each other. We distinguish two kinds of figure relations, which are crucial for
+annotation: non-referential and referential ones. Referential relations include constellations in which the body of
+one and the same figure is represented several times but in different ways. In addition to shadows (Fig. 5e), these
+mainly include reflections, e.g., in mirrors (Fig. 5f), in water (Fig. 5g), and on surfaces like metallic armor. We set
+the corresponding bounding boxes whenever the referencing part, the reflection, can be recognized as human-like
+even without the referenced part, i.e., the human reflected in some way. Non-referential relations are found
+when figures overlap, intersect, or are symmetrically arranged (Fig. 5h). In case of overlaps and intersections, we
+approximate occluded keypoints as far as possible.
+Deviations from the ‘ideal’ human body. The ideal human body has been studied since antiquity [25, 62, 68, 72]:
+from scholars like Vitruvius [92], to medieval draftsmen such as Villard de Honnecourt [31], Renaissance artists
+Leonardo da Vinci [38, 39, 58] and Albrecht Dürer [32, 65], or even modernists like Oskar Schlemmer [45].
+We declare bodies as deviating from this ideal whose depicted measurements or proportions do not adhere to
+usual conventions. Three subcategories are discerned. (a) Often deviations are due to stylistic reasons expressed
+regionally, epochally, or individually. The lack of differentiation of the entire body shape or individual body parts
+is characteristic of Impressionism and Pointillism (Fig. 5i); figures veiled by robes that fundamentally obscure the
+body are common in Art Nouveau as well as Japanese woodblock prints of the Ukiyo-e [91]. If the placement of
+keypoints in an image is complicated by blurred contours, distorted proportions, or missing joints, as shown
+in Fig. 5j, we approximate them, provided the figures can be recognized as human. (b) Another subcategory
+comprises non-human bodies and body parts. These include mythological figures such as centaurs, harpies, and
+mermaids (Fig. 5k), biblical figures, e.g., angels, and human-like animals like monkeys and lemurs. While animals
+are excluded from annotation, we annotate human parts of mythological and biblical figures; consequently, the
+animal limbs of centaurs are not annotated, nor are the wings and halo of angels. (c) The third subcategory
+covers non-living bodies and body parts, with a considerable portion being severed heads (Fig. 5l)7 and limbs.
+The latter may again result from the analog original itself, for instance, as part of the composition, but may also
+be grounded in the digital reproduction, e.g., in particularly detailed views or images in need of restoration that
+no longer permit keypoints to be fully labeled. While severed limbs are not annotated, severed heads are, since
+they generally allow for more keypoints and constitute a more substantial part of the human body than hands
+or legs. Also included are fabricated bodies, such as dolls, masks, crafts, sculptures, and images within images
+depicting human bodies, e.g., in salon paintings.
+Positioning of the human body. Of relevance is the body’s positioning in the image space especially for back
+views, as in Dürer’s Hercules (1498; Fig. 5m), where the inversion of keypoints must be taken into account. For
+profile views, it is crucial to set the eyes and ears on the non-visible side of the face as well. This applies, e.g., to
+Florentine portraits (Fig. 5n), which refer to the strict profile of emperors on ancient coins [17]. In a considerable
+number of images, perspective distortions are furthermore present, along with twists and turns. They are found
+primarily in Baroque and Rococo works such as those by Tiepolo (Fig. 5o), but also in 20th-century avant-garde
+7See iconographies such as David and Goliath, Judith and Holofernes, and Salomé and John the Baptist.
+
+Poses of People in Art
+•
+11
+Table 3. The People-Art [84] and PoPArt data sets are descriptively compared. Figures are indicated by bounding boxes
+associated with them. Up to 17 keypoints are stored per figure. Difficult-to-annotate figures are included.
+Data set
+Split
+Images
+Images𝑃𝑜𝑠
+Images𝑁𝑒𝑔
+Figures
+Crowds
+Keypoints
+Styles
+People-Art
+Training
+1,623
+525
+1,098
+1,512
+0
+0
+43
+Validation
+1,383
+442
+941
+1,219
+0
+0
+43
+Testing
+1,616
+522
+1,094
+1,137
+0
+0
+43
+Total
+4,622
+1,489
+3,133
+3,868
+0
+0
+43
+PoPArt
+Training
+1,472
+1,472
+0
+6,457
+245
+33,582
+22
+Validation
+491
+491
+0
+2,175
+114
+11,104
+22
+Testing
+491
+491
+0
+2,117
+106
+11,468
+22
+Total
+2,454
+2,454
+0
+10,749
+465
+56,154
+22
+movements, as in the French surrealist André Masson (Fig. 5p). There, too, the possible inversion of keypoints
+has to be considered. If a figure is twisted to such an extent that keypoints cannot be approximated, individual
+limbs are omitted and annotated only up to the last keypoint visible or to be approximated.
+3.2.3
+Accuracy. Figure instance annotations were checked in several test cycles according to formerly stated
+guidelines. They were once again reviewed at the end of the annotation process. We verified, e.g., that each
+keypoint referred to the correct body part, that body halves were properly labeled, especially for back views and
+twisted figures, and that bounding boxes surrounded only the extent of the figure visible in the image.
+3.3
+Descriptive Statistics
+For machine learning purposes, the PoPArt data set is divided into three subsets: training, validation, and testing.
+They contain 1,472, 491, and 491 images, respectively, so approximate split ratios of 60 %, 20 %, and 20 % are met.
+In contrast to the People-Art data set [84], we do not reduce image sizes to a maximum scale of 500 × 500 pixels,
+but directly redistribute the digital reproductions from WikiArt. The widest image measures 6,298 × 3,049 and the
+highest 4,524 × 6,018 pixels. Figure instance annotations total 6,457 in PoPArt for training, 2,175 for validation,
+and 2,117 for testing, with keypoint annotations of 33,582, 11,104, and 11,468, respectively. To maintain an equal
+distribution of figure instances across data splits, we applied the following procedure: images were first grouped
+by depiction style and then sorted in descending order based on the number of figure instance annotations.
+Considering the split ratios, we then processed batches of five images and randomly assigned three of them as
+training samples, one as validation, and one as testing.
+Table 3 summarizes PoPArt in comparison to the similarly constituted People-Art data set. Both data sets
+focus exclusively on human figures depicted in art-historical objects; other classes are not annotated. While
+People-Art’s core application solely lies in the computer-aided detection of figures, PoPArt is designed to support
+both their detection and that of their keypoints. Further structural differences arise. (i) People-Art contains more
+training, validation, and testing images due to the integration of negative image samples that do not show human
+figures. However, PoPArt features almost three times as many positive samples in the training set, which have
+at least one instance annotation. This includes more small-area instances measuring between 0 and 162 pixels,
+namely 0.9 %. In the People-Art data set, it amounts to only 0.5 % despite reduced image sizes. In addition, the
+data set completely lacks crowd annotations. PoPArt thus decisively enables the automatic detection even of
+figures that are displayed small. (ii) Moreover, PoPArt accounts for the broad spectrum of art-historical body
+language in two ways. As shown in Fig. 6, the proportion of images with at least six figures is 8.52 % higher in
+
+12
+•
+S. Schneider and R. Vollmer
+0%
+25%
+50%
+75%
+100%
+1
+2
+3
+4
+5
+> 5
+Number of figures
+Percentage of images
+(a) People-Art
+0%
+25%
+50%
+75%
+100%
+1
+2
+3
+4
+5
+> 5
+Number of figures
+Percentage of images
+(b) PoPArt
+Fig. 6. The proportion of images with at least six figures is 8.52 % higher in the PoPArt than in the People-Art data set [84].
+This is also reflected in a larger maximum number of figures in an image: it is 28 for People-Art and 110 for PoPArt.
+0%
+10%
+20%
+30%
+40%
+50%
+0
+1
+2
+3
+4
+5
+> 5
+Number of overlaps
+Percentage of instances
+(a) People-Art
+0%
+10%
+20%
+30%
+40%
+50%
+0
+1
+2
+3
+4
+5
+> 5
+Number of overlaps
+Percentage of instances
+(b) PoPArt
+Fig. 7. The PoPArt data set has a 14.22 % higher share of overlapping figure instances than the People-Art data set [84], with
+the maximum number of overlaps in an image being 11 for People-Art and 23 for PoPArt.
+the PoPArt than in the People-Art data set. This is reflected in a larger maximum number of figures in an image:
+it is 28 for People-Art and 110 for PoPArt. As a result, the PoPArt data set also has a 14.22 % higher share of
+overlapping figure instances than People-Art (Fig. 7), with the maximum number of overlaps in an image being
+11 for People-Art and 23 for PoPArt.
+3.4
+Data Split Format
+All data splits follow the JSON-based Microsoft COCO format [47]; Fig. 8 displays an exemplary figure instance
+annotation with its referencing image annotation. For each image annotation, we provide metadata (wikiart_url,
+wikiart_image_url, and wikiart_style) in addition to mandatory fields (id, license, width, height, and
+file_name). Figure instance annotations contain task-agnostic information (id, image_id, and category_id),
+supplemented by fields essential to the respective detection task. The fields bbox, segmentation, and iscrowd
+are declared for both figure instance and keypoint detection, while keypoints and num_keypoints are noted for
+keypoint detection only. Through a 53-dimensional array, each of the 17 keypoints is represented with three
+values: its location, 𝑥 and 𝑦, and a visibility flag 𝑣 that indicates whether the respective keypoint is visible and
+labeled, 𝑣 = 2, or not, 𝑣 = 0. In contrast to the Microsoft COCO format guidelines, we assign 𝑣 = 2 to index
+occluded keypoints as well, rather than 𝑣 = 1. Keypoints are recorded in the order established by the COCO
+
+Poses of People in Art
+•
+13
+{
+· · ·
+"images": [
+· · ·
+{
+"id": 58,
+"license": 1,
+"width": 2310,
+"height": 3000,
+"file_name": "albrecht -durer_death -of -orpheus -1498. jpg",
+"metadata": {
+"wikiart_url": "https ://www.wikiart.org/en/albrecht -durer/death -of -orpheus -1498" ,
+"wikiart_image_url": "https :// uploads.wikiart.org/images/albrecht -durer/death -of -orpheus
+-1498. jpg",
+"wikiart_style": "Northern Renaissance"
+}
+},
+· · ·
+],
+"annotations": [
+· · ·
+{
+"id": 64,
+"image_id": 58,
+"category_id": 1,
+"area": 1319355,
+"bbox": [ 616.0, 1718.0, 1305.0, 1011.0 ],
+"segmentation": [ [ 1921.0, 1718.0, 1921.0, 2729.0, 616.0, 2729.0, 616.0, 1718.0 ] ],
+"keypoints": [ 950, 1872, 2, 945, 1848, 2, 904, 1870, 2, 924, 1873, 2, 864, 1918, 2, 1108,
+1905, 2, 871, 2100, 2, 1279, 1778, 2, 790, 2339, 2, 1085, 1789, 2, 750, 2620, 2, 1313,
+2311, 2, 1147, 2339, 2, 1600, 2567, 2, 902, 2629, 2, 1870, 2456, 2, 1200, 2431, 2 ],
+"num_keypoints": 17,
+"iscrowd": false
+},
+· · ·
+]
+}
+Fig. 8. PoPArt follows the JSON-based Microsoft COCO format [47], for which a figure instance annotation with its
+referencing image annotation is displayed.
+format: nose, left and right eye, left and right ear, left and right shoulder, left and right elbow, left and right wrist,
+left and right hip, left and right knee, left and right ankle.
+4
+APPLICATIONS
+In the course of this section, we consider application scenarios in which PoPArt can be usefully integrated and,
+building on these, discuss prospects for a digitally supported art history. Two scenarios are distinguished: those
+arising from human pose estimation (Section 4.1), and those from human figure detection (Section 4.2).
+4.1
+Human Pose Estimation
+In a first application scenario, we demonstrate that PoPArt enables the quantitatively systematized exploration
+of human poses in visual art. For this purpose, we suggested in Springstein et al. [73] a two-stage approach
+based on two Transformer models [14, 79]: the first model detects bounding boxes of human figures, while the
+second one analyzes the individual boxes for keypoints (Fig. 9). We in this context adapted a semi-supervised
+
+14
+•
+S. Schneider and R. Vollmer
+Set of Query Embeddings
+Set of Query Embeddings
+Positional Encoding 
+Positional Encoding
+Set of Keypoints
+Set of Boxes
+CNN Backbone
+Transformer Encoder
+Transformer Decoder
+Crop
+CNN Backbone
+Transformer Encoder
+Transformer Decoder
+Fig. 9. The two-stage human pose estimator from Springstein et al. [73] uses two Transformer models: the input of the first
+stage is the entire image, for which the first Transformer predicts a fixed set of bounding boxes. The individual boxes are
+cropped and serve as input for the second stage; the second Transformer model then computes a set of keypoints.
+(a) Default image grid
+(b) Two-dimensional canvas view
+Fig. 10. With the aid of the web platform iART [67, 74], the process of comparative vision is facilitated by various object
+views, as illustrated by the example of Fall of Man.
+learning technique to reduce the performance loss caused by the shift between existing real-world data sets and
+the art-historical domain, and to reduce the quantity of domain-specific annotation data. The basic principle is
+to use both labeled and unlabeled image material to train a student model. The teacher serves as a generator
+of pseudo-labels; to this end, unlabeled images are first weakly augmented and then used for the detection of
+human figures, just as figures enclosed by bounding boxes are weakly augmented and used to predict their
+keypoints. Three art-historical data sets are plugged into the routine: in addition to PoPArt, we also employ
+People-Art [84] for labeled and Art500k [50] for unlabeled data. Experiments performed on the PoPArt test set
+in comparison to more established approaches that apply pre-trained models [35, 48] or enrich real-world data
+sets with style transfer [49] indicated that the performance of human pose estimators is greatly enhanced by
+using semi-supervised methods with additional unlabeled data. Moreover, in a user study, we also confirmed the
+feasibility of the approach for retrieval tasks, enabling the search for resembling poses. The pose—as the holistic
+
+@iART
++  adam and eve &
+XEN 
+① Global Weights
+器 Result View@iART
+Q曾 +日 adam and eve α
+XEN 
+① Global Weights
+品 Result View
+I Cluster DisplayPoses of People in Art
+•
+15
+Table 4. Figure detection results are reported for the People-Art test set [84]. For training and validation, PoPArt was used
+in addition to People-Art. In contrast to previous benchmarks by Kadish et al. [36] and Gonthier et al. [30], we include
+difficult-to-annotate figures. The best performing approach is indicated in bold.
+Model
+Backbone
+LR
+AP
+AP50
+AP75
+AP𝑆
+AP𝑀
+AP𝐿
+AR
+TOOD [22]
+ResNet-50-FPN
+2e − 4
+0.478
+0.780
+0.499
+0.162
+0.311
+0.511
+0.654
+PVT [81]
+PVTv2-B2
+1e − 5
+0.497
+0.805
+0.518
+0.076
+0.315
+0.532
+0.625
+Cascade R-CNN [11]
+ResNet-50-FPN
+2e − 4
+0.464
+0.761
+0.490
+0.152
+0.307
+0.495
+0.606
+SABL Cascade R-CNN [80]
+ResNet-50-FPN
+2e − 4
+0.456
+0.762
+0.457
+0.116
+0.311
+0.487
+0.601
+Faster R-CNN [63]
+ResNet-50-FPN
+2e − 4
+0.439
+0.770
+0.447
+0.128
+0.312
+0.465
+0.580
+SABL Faster R-CNN [80]
+ResNet-50-FPN
+2e − 4
+0.453
+0.756
+0.463
+0.129
+0.308
+0.483
+0.604
+PISA Faster R-CNN [13]
+ResNet-50-FPN
+2e − 4
+0.447
+0.767
+0.464
+0.133
+0.306
+0.475
+0.582
+Libra Faster R-CNN [56]
+ResNet-50-FPN
+2e − 4
+0.442
+0.769
+0.451
+0.084
+0.312
+0.471
+0.583
+abstraction of bodily expression—can thus prove elemental to the formulaic recapitulation of significant motifs
+through computational assistance.
+This becomes particularly evident when machine-generated similarity arrangements are explored through
+web-based user interfaces. For instance, on the platform iART [67, 74],8 object retrieval is performed not only
+based on art-historical keywords generated by deep learning, but also by leveraging state-of-the-art multimodal
+embeddings such as the Transformer-backed neural network CLIP, which creates a unified feature space for image
+and text [60]. First, the retrieval of certain iconographies is thereby enabled. As illustrated in Fig. 10a, searching
+for “adam and eve” primarily returns the classical Renaissance depiction of the Fall of Man, in which Adam and
+Eve stand image-parallel, left and right under the Tree of Knowledge. The iconography can be examined more
+in-depth if, on top of CLIP-based pre-filtering, the pose embeddings of each figure are determined and then
+mapped onto a two-dimensional canvas using the dimensionality reduction technique UMAP [51]. Several cluster
+structures emerge in Fig. 10b: the one shown at the top left, e.g., reveals an image group of more dynamic poses
+that are conspicuous for their bent or flared legs; apart from the fact that here the apple is being handed to Adam
+in a rather prominent manner.
+4.2
+Human Figure Detection
+We show in the second application scenario that as a by-product of PoPArt’s domain-specific curation, the
+sole detection of art-historical figures is decisively improved. For this purpose, we utilize the same models and
+pipeline as described in Section 3.1 for the preliminary step of our semi-automatic image collection procedure:
+models are first pre-trained on Microsoft COCO 2017 for 12 epochs and then fine-tuned, with their classification
+head re-initialized, for another 12 epochs—now on both the People-Art [84] and PoPArt data sets. Parameter
+settings remain unchanged; the learning rate is, again, set to 2e − 4 in case of ResNet-50 and 1e − 5 in case of
+Transformer backbones. Compared to those in Table 2, the benchmarks shown in Table 4 clearly demonstrate
+that AP and AR increase considerably for all models when PoPArt is integrated into the training routine. For the
+Transformer-based PVT model [81], e.g., AP and AR improve to the same extent, from 46.5 to 49.7 % and 60.1
+to 62.5 %, respectively. The leap is even more noticeable if we plug-in the PoPArt instead of the People-Art test
+set. AP then rises from 36.6 to 43.6 % and AR from 48.2 to 55.0 % for PVT. At the same time, this reconfirms the
+greater complexity of the figures contained in PoPArt, which are exhaustively marked in the images by bounding
+boxes, even if they are very small or appear in crowds, and hence overlap frequently. The additional integration
+8https://www.iart.vision/.
+
+16
+•
+S. Schneider and R. Vollmer
+(a) Nicholas Poussin, The Deluge (1660–1664)
+(b) John Martin, The Deluge (1834)
+Fig. 11. Detail views of the crowds depicted in Nicholas Poussin’s and John Martin’s versions of The Deluge, respectively.
+Both images have been slightly lightened to emphasize depiction specifics. The images are in the public domain.
+of PoPArt into the training routine thus is particularly advantageous to movements that emphasize the depiction
+of a larger number of people, as in Mannerism and the regional expressions of the Renaissance; in Northern
+Renaissance works, e.g., AP improves from 27.1 to 33.5 % and AR from 37.5 to 43.4 % (Table 5 in Appendix).
+Indeed, this image of the crowd, from small gatherings in village squares to streams of passers-by in modern
+pedestrian zones, benefits especially from computer-aided methods of detection; even if these may initially only
+be used to pre-filter the (digitally available) image material. Namely, the crowd’s underlying constitution, which
+has been increasingly received since the 18th century [40], becomes strictly quantifiable: by the number of people
+in it, their proximity or distance from each other, the space they occupy in the image, and in relation to other
+subjects. John Martin’s emphatically apocalyptic Deluge (1834; Fig. 11b), for instance, focuses on the entirely
+de-individualized crowd—a multitude of people depicted in a confined space, who are “tossed back and forth like
+cue balls” [44]. In Nicholas Poussin’s Deluge (1660–1664; Fig. 11a), on the other hand, the majority of figures
+are still, because of the larger body size, differentiated in their moments of action. While a man clings to his
+horse in the foreground, a mother, slightly moved back, stretches her child upwards to the shore. It is precisely
+these iconographic traditions that first become easily decipherable in larger amounts of data through distant
+viewing and only then are examined in detail from a more art-historical perspective. Recommender systems like
+iART [67, 74] can ultimately point to research-worthy phenomena here as well.
+5
+CONCLUSION
+In this paper, we introduced with Poses of People in Art the first publicly available and openly licensed data set for
+estimating human poses in visual art. It consists of 2,454 images from 22 art-historical depiction styles, including
+those that increasingly turned away from lifelike representations of the body and toward artificial forms. A
+total of 10,749 human figures are enclosed by rectangular bounding boxes, with a maximum of four per image
+labeled by up to 17 keypoints. For machine learning purposes, the data set is pre-split into three subsets—training,
+validation, and testing—, each following the JSON-based Microsoft COCO format. In addition to mandatory fields,
+image annotations provide metadata from the art-historical online encyclopedia WikiArt. As illustrated in two
+application scenarios, the data set not only validates the performance of deep-learning models, but in this way
+enables the comprehensive investigation of body phenomena in art—whether at the level of individual figures,
+whose bodily subtleties are captured, or entire figure constellations, whose position, distance, or proximity to
+one another is considered. With the further aid of readily accessible online platforms like the presented iART, we
+see the potential to reveal large-scale disruptions of formal conventions and make them interactively explorable.
+
+Poses of People in Art
+•
+17
+Since this would allow hitherto marginalized collections to be easily included in analyses, the discipline of art
+history would benefit from an increasingly de-canonized gaze that is no longer primarily devoted to European
+art. Intra- as well as inter-iconographic recurrent motifs, whose radically altered semantics are disconcerting,
+might be thoroughly discussed for the first time in this context.
+ACKNOWLEDGMENTS
+This work was funded in part by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG)
+under project no. 415796915. We thank Ursula Huber for her valuable support with the image annotation. We
+also thank Hubertus Kohle, Ralph Ewerth, and Matthias Springstein for fruitful discussions and useful comments
+on the subject matters.
+AUTHORS’ CONTRIBUTIONS
+S.S. conceived, designed, and performed the experiments, analyzed the data, oversaw image annotation, and
+ensured data quality; R.V. performed image annotation. S.S. and R.V. wrote the manuscript, and read, commented,
+and approved the final version.
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+[84] Nicholas Westlake, Hongping Cai, and Peter Hall. 2016. Detecting People in Artwork with CNNs. In Computer Vision – ECCV 2016
+Workshops (Lecture Notes in Computer Science, Vol. 9913). Springer, Cham, 825–841. https://doi.org/10.1007/978-3-319-46604-0_57
+[85] WikiArt 2010. WikiArt.org. Visual Art Encyclopedia. Retrieved January 13, 2023 from https://www.wikiart.org/
+[86] Michael J. Wilber, Chen Fang, Hailin Jin, Aaron Hertzmann, John P. Collomosse, and Serge J. Belongie. 2017. BAM! The Behance
+Artistic Media Dataset for Recognition Beyond Photography. In IEEE/CVF International Conference on Computer Vision. IEEE, New York,
+1211–1220. https://doi.org/10.1109/ICCV.2017.136
+[87] John Willats. 1997. Art and Representation. New Principles in the Analysis of Pictures. Princeton University Press, Princeton.
+[88] Victoria Yanulevskaya, Jasper R. R. Uijlings, Elia Bruni, Andreza Sartori, Elisa Zamboni, Francesca Bacci, David Melcher, and Nicu Sebe.
+2012. In the Eye of the Beholder. Employing Statistical Analysis and Eye Tracking for Analyzing Abstract Paintings. In MM ’12: The 20th
+ACM International Conference on Multimedia, Noboru Babaguchi, Kiyoharu Aizawa, John R. Smith, Shin’ichi Satoh, Thomas Plagemann,
+Xian-Sheng Hua, and Rong Yan (Eds.). ACM, New York, 349–358. https://doi.org/10.1145/2393347.2393399
+[89] Pradeep Yarlagadda, Antonio Monroy, Bernd Carqué, and Björn Ommer. 2010. Recognition and Analysis of Objects in Medieval Images.
+In Computer Vision – ECCV 2010 Workshops (Lecture Notes in Computer Science, Vol. 6469), Reinhard Koch and Fay Huang (Eds.). Springer,
+Cham, 296–305. https://doi.org/10.1007/978-3-642-22819-3_30
+[90] Nikolaos-Antonios Ypsilantis, Noa Garcia, Guangxing Han, Sarah Ibrahimi, Nanne van Noord, and Giorgos Tolias. 2021. The Met
+Dataset. Instance-level Recognition for Artworks. In Proceedings of the Neural Information Processing Systems Track on Datasets and
+Benchmarks 1, NeurIPS Datasets and Benchmarks 2021, Joaquin Vanschoren and Sai-Kit Yeung (Eds.). 12 pages. Retrieved January 13,
+2023 from https://datasets-benchmarks-proceedings.neurips.cc/paper/2021/file/5f93f983524def3dca464469d2cf9f3e-Paper-round2.pdf
+[91] Linda Gertner Zatlin. 1997. Beardsley, Japonisme and the Perversion of the Victorian Ideal. Cambridge University Press, Cambridge.
+[92] Frank Zöllner. 2004. Anthropomorphismus. Das Maß des Menschen in der Architektur von Vitruv bis Le Corbusier. In Ist der Mensch
+das Maß aller Dinge? Beiträge zur Aktualität des Protagoras, Otto Neumaier (Ed.). Bibliopolis, Möhnesee, 306–344.
+APPENDIX
+Table 5. Figure detection results are reported for the PoPArt test set by depiction style and training set(s); TOOD [22] is
+employed as model, respectively. In contrast to previous benchmarks by Kadish et al. [36] and Gonthier et al. [30], we include
+difficult-to-annotate figures.
+Style
+Training Set(s)
+AP
+AP50
+AP75
+AP𝑆
+AP𝑀
+AP𝐿
+AR
+Abstract Expressionism
+People-Art
+0.900
+1.000
+1.000
+0.900
+0.900
+People-Art, PoPArt
+0.850
+1.000
+1.000
+0.850
+0.850
+Art Nouveau
+People-Art
+0.425
+0.677
+0.441
+0.228
+0.449
+0.643
+People-Art, PoPArt
+0.460
+0.762
+0.452
+0.222
+0.486
+0.722
+Baroque
+People-Art
+0.301
+0.461
+0.322
+0.000
+0.024
+0.444
+0.386
+People-Art, PoPArt
+0.357
+0.539
+0.389
+0.051
+0.047
+0.512
+0.498
+Contemporary Realism
+People-Art
+0.624
+0.864
+0.724
+0.316
+0.501
+0.730
+0.720
+People-Art, PoPArt
+0.627
+0.851
+0.746
+0.255
+0.632
+0.755
+0.729
+Cubism
+People-Art
+0.511
+0.836
+0.565
+0.750
+0.511
+0.696
+People-Art, PoPArt
+0.601
+0.876
+0.655
+0.676
+0.607
+0.752
+Early Renaissance
+People-Art
+0.427
+0.696
+0.442
+0.000
+0.178
+0.541
+0.563
+People-Art, PoPArt
+0.503
+0.774
+0.548
+0.000
+0.295
+0.605
+0.629
+Expressionism
+People-Art
+0.567
+0.832
+0.594
+0.486
+0.627
+0.711
+
+22
+•
+S. Schneider and R. Vollmer
+Table 5. Figure detection results are reported for the PoPArt test set by depiction style and training set(s); TOOD [22] is
+employed as model, respectively. In contrast to previous benchmarks by Kadish et al. [36] and Gonthier et al. [30], we include
+difficult-to-annotate figures.
+Style
+Training Set(s)
+AP
+AP50
+AP75
+AP𝑆
+AP𝑀
+AP𝐿
+AR
+People-Art, PoPArt
+0.592
+0.839
+0.627
+0.474
+0.667
+0.754
+Fauvism
+People-Art
+0.493
+0.765
+0.534
+0.126
+0.579
+0.622
+People-Art, PoPArt
+0.576
+0.845
+0.643
+0.171
+0.657
+0.690
+High Renaissance
+People-Art
+0.254
+0.405
+0.268
+0.000
+0.052
+0.424
+0.340
+People-Art, PoPArt
+0.310
+0.481
+0.319
+0.002
+0.068
+0.514
+0.417
+Impressionism
+People-Art
+0.473
+0.737
+0.489
+0.000
+0.341
+0.520
+0.616
+People-Art, PoPArt
+0.509
+0.777
+0.527
+0.000
+0.397
+0.549
+0.656
+Mannerism
+People-Art
+0.298
+0.540
+0.283
+0.000
+0.069
+0.397
+0.483
+People-Art, PoPArt
+0.371
+0.668
+0.343
+0.022
+0.192
+0.459
+0.542
+Naive Art
+People-Art
+0.291
+0.470
+0.304
+0.166
+0.150
+0.396
+0.443
+People-Art, PoPArt
+0.394
+0.679
+0.379
+0.175
+0.279
+0.487
+0.528
+New Realism
+People-Art
+0.514
+0.803
+0.538
+0.557
+0.521
+0.665
+People-Art, PoPArt
+0.553
+0.842
+0.547
+0.373
+0.593
+0.716
+Northern Renaissance
+People-Art
+0.271
+0.460
+0.286
+0.015
+0.128
+0.410
+0.375
+People-Art, PoPArt
+0.335
+0.555
+0.350
+0.052
+0.196
+0.481
+0.434
+Pointillism
+People-Art
+0.465
+0.726
+0.556
+0.000
+0.467
+0.519
+0.567
+People-Art, PoPArt
+0.553
+0.827
+0.644
+0.010
+0.570
+0.600
+0.638
+Pop Art
+People-Art
+0.454
+0.628
+0.499
+0.041
+0.258
+0.555
+0.559
+People-Art, PoPArt
+0.514
+0.683
+0.580
+0.164
+0.351
+0.600
+0.667
+Post Impressionism
+People-Art
+0.607
+0.903
+0.660
+0.396
+0.628
+0.732
+People-Art, PoPArt
+0.672
+0.911
+0.704
+0.445
+0.693
+0.768
+Realism
+People-Art
+0.657
+0.869
+0.768
+0.000
+0.047
+0.727
+0.746
+People-Art, PoPArt
+0.693
+0.915
+0.721
+0.030
+0.347
+0.756
+0.787
+Rococo
+People-Art
+0.534
+0.787
+0.595
+0.023
+0.601
+0.629
+People-Art, PoPArt
+0.606
+0.866
+0.654
+0.184
+0.654
+0.710
+Romanticism
+People-Art
+0.303
+0.499
+0.306
+0.000
+0.098
+0.436
+0.414
+People-Art, PoPArt
+0.401
+0.619
+0.440
+0.000
+0.176
+0.557
+0.508
+Symbolism
+People-Art
+0.322
+0.574
+0.316
+0.068
+0.228
+0.360
+0.458
+People-Art, PoPArt
+0.362
+0.674
+0.362
+0.069
+0.276
+0.403
+0.521
+Ukiyo-e
+People-Art
+0.414
+0.746
+0.437
+0.000
+0.050
+0.460
+0.619
+People-Art, PoPArt
+0.437
+0.830
+0.429
+0.009
+0.080
+0.479
+0.638
+

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+arXiv:2301.04256v1  [hep-th]  11 Jan 2023
+Quantum backreaction for overspinning BTZ geometries
+Olaf Baake2,1 ∗ and Jorge Zanelli1,3 †
+1Centro de Estudios Científicos (CECs), Arturo Prat 514, Valdivia, Chile
+2Instituto de Matemáticas, Universidad de Talca, Casilla 747, Talca 3460000, Chile
+3Universidad San Sebastián, General Lagos 1163, Valdivia, Chile
+January 12, 2023
+Abstract
+We examine the semiclassical backreaction of a conformally coupled scalar field on an over-
+spinning BTZ geometry. This extends the work done on a similar problem for (2 + 1)- AdS
+geometries of the BTZ family with |M| > |J|. The overspinning classical solutions corresponds
+to |M| < |J| and possess a naked singularity at r = 0.
+Using the renormalized quantum
+stress-energy tensor for a conformally coupled scalar field on such a spacetime, we obtain the
+semiclassical Einstein equations, which we attempt to solve perturbatively. We show that the
+stress-energy tensor is non-renormalizable in this approach, and consequently the perturbative
+solution to the semiclassical equations in the overspinning case does not exist. This could be an
+indication of the fact that the naked singularity at the center of an overspinning geometry is of
+a more severe nature than the conical singularity found in the same family of BTZ geometries.
+1
+Introduction
+Since the dawn of general relativity, many black hole solutions to Einstein’s field equations have been
+found. All these black holes contain a spacetime singularity hidden by an event horizon. However,
+for some range of values of the integration constants (mass M, angular momentum J, electric charge
+Q) these solutions have no event horizon. Although paradoxical, these naked singularities are exact
+solutions to the classical equations of general relativity as well. In the vicinity of a naked singularity
+causality and other physical laws can be arbitrarily violated, which is why Roger Penrose suggested
+the existence of a (weak) cosmic censorship principle in nature [1], requiring singularities to be
+hidden behind an event horizon. In that case, an outside observer would be causally disconnection
+from the singularity.
+Classically, naked singularities cannot be ruled out on mathematical grounds, and it is difficult to
+prove that every possible collapse process leads to the formation of an event horizon. The fact that
+so far no naked singularities have been observed in the universe may be interpreted as an indication
+that, in the strong gravity regime near a singularity, quantum gravity effects dominate eliminating
+singularities altogether, or at least making sure that a horizon forms around them.
+The accumulation of experiments and observations that confirm the predictions of general rel-
+ativity puts very tight constraints on possible theories incorporating both general relativity and
+∗[email protected]
+†[email protected]
+1
+
+quantum theory. Since both theories are so well established in their regimes, it is sensible to look
+for a common area where a semi-classical approach could be used to obtain a better understanding
+of the issues at hand. Calculating quantum effects on a curved background spacetime is notoriously
+difficult, but in (2+1)-dimensional AdS spacetime this problem becomes significantly simpler and
+still provide meaningful information to learn from.
+The Bañados-Teitelboim-Zanelli (BTZ) black hole in (2+1)-dimensional AdS spacetime [2, 3],
+obtained for M ≥ |J| are particularly interesting geometries in this respect, but these are not
+the only solutions of physical interest in this theory and with the same global symmetries.
+Lo-
+cally constant curvature 2+1 spacetimes include, besides the BTZ black hole family, the self-dual
+Coussaert-Henneaux spacetimes [4], and the toroidal time-dependent geometries [5], with global
+isometry groups SO(2) × R SO(2) × SO(2, 1) and SO(2) × SO(2), respectively.
+Recently, the quantum back reaction on the classical singularities was studied for several geome-
+tries, including static, rotating and extremal BTZ black holes, as well as for static and rotating
+conical naked singularities [6, 7, 8, 9]. The naked singularities considered in these papers are contin-
+uations of the BTZ spacetime to the case of negative mass [10]. The interesting aspect of this result
+is that the quantum fluctuations of a conformally coupled scalar field generate a non-vanishing stress
+energy-momentum tensor that through Einstein’s equations produces aback-reacted geometry with
+a horizon of order Planck length in radius. This dressing up of the naked singularity, turning it into
+a black hole, could be viewed as a mechanism that implements cosmic censorship. These results
+have also been confirmed by an alternative holographic approach in [11].
+Here we are concerned with the overspinning BTZ spacetime, which occurs if the absolute value
+of the angular momentum is greater than that of the mass. This geometry is also endowed with a
+naked singularity at r = 0, as in the case of the conical singularity obtained for M ≤ −|J|.
+We show that the stress-energy tensor contains incurable divergences, making the perturbative
+ansatz to the semiclassical equations of motion ill-defined. While the equations of motion can still be
+formally integrated, the first order corrections to the metric functions would become large, further
+demonstrating the inapplicability of a perturbative approach to this type of geometry. This strongly
+suggests that the naked singularity of an overspinning geometry is of a more severe nature than
+the conical singularities appearing in the other BTZ geometries so that they cannot be cured by a
+perturbative quantum censor.
+2
+Overspinning BTZ space-time
+The rotating BTZ metric [2, 3], is given by
+ds2 = −
+�r2
+l2 − M
+�
+dt2 − Jdtdθ +
+�r2
+l2 − M + J2
+4r2
+�−1
+dr2 + r2dθ2,
+(1)
+where the coordinate ranges are: −∞ < t < ∞, 0 < r < ∞ and 0 ≤ θ < 2π, Λ = −l−2 is the
+cosmological constant, and M and J are mass and angular momentum respectively. This metric
+describes different spacetimes that can be classified by the values of M and J which determine the
+nature of the four roots of the equation grr = 0,
+λ± = l
+2
+��
+M + J
+l ±
+�
+M − J
+l
+�
+.
+(2)
+These roots are real for M ≥ |J|/l (black holes) and take complex values for M < |J|/l (naked
+singularities). The full classification is explained in detail in [3], but here we will consider the so-
+called overspinning geometry (|M|l < |J|). This geometry was examined in [12] through the study
+2
+
+of classical geodesics around it. In particular, we will analyze the back reaction of the geometry to
+the presence of a conformally coupled quantum scalar field, following the steps in [6, 7, 8, 9], where
+the back reaction for conical naked singularities in the parameter range M ≤ −|J| was studied.
+The starting point of the analysis is the observation that the BTZ spacetimes (1) are quotients of
+the universal covering of anti-de Sitter space-time (CAdS3) by an appropriate Killing vector field [3].
+The constant negative curvature spacetime AdS3 is defined by a pseudosphere of radius l embedded
+in R(2,2) as
+ηABXAXB = −
+�
+X0�2 +
+�
+X1�2 +
+�
+X2�2 −
+�
+X3�2 = −l2 .
+(3)
+The metric reads
+ηABdXAdXB = −
+�
+dX0�2 +
+�
+dX1�2 +
+�
+dX2�2 −
+�
+dX3�2 ,
+(4)
+where the embedding coordinates XA must be specified as functions of (t, r, θ). As shown in [12],
+the overspinning geometry (1) with |M| < |J| corresponds to embedding coordinates given by
+X0 = l
+2
+√
+A + 1 cosh [a (t/l − θ)] {cos [b (θ + t/l)] − sin [b (θ + t/l)]}
++ǫ l
+2
+√
+A − 1 sinh [a (t/l − θ)] {sin [b (θ + t/l)] + cos [b (θ + t/l)]} ,
+(5)
+X1 = l
+2
+√
+A + 1 sinh [a (t/l − θ)] {cos [b (θ + t/l)] − sin [b (θ + t/l)]}
++ǫ l
+2
+√
+A − 1 cosh [a (t/l − θ)] {sin [b (θ + t/l)] + cos [b (θ + t/l)]} ,
+(6)
+X2 = l
+2
+√
+A + 1 sinh [a (t/l − θ)] {sin [b (θ + t/l)] + cos [b (θ + t/l)]}
+−ǫ l
+2
+√
+A − 1 cosh [a (t/l − θ)] {cos [b (θ + t/l)] − sin [b (θ + t/l)]} ,
+(7)
+X3 = l
+2
+√
+A + 1 cosh [a (t/l − θ)] {sin [b (θ + t/l)] + cos [b (θ + t/l)]}
+−ǫ l
+2
+√
+A − 1 sinh [a (t/l − θ)] {cos [b (θ + t/l)] − sin [b (θ + t/l)]} ,
+(8)
+where
+a =
+�
+|J|/l + M
+2
+,
+b =
+�
+|J|/l − M
+2
+,
+A =
+2
+�
+J2
+4 + r4
+l2 − Mr2
+√
+J2 − l2M 2
+,
+(9)
+with ǫ = sign(M − r2/l2). Note that both cases (ǫ = ±1) lead to the same RSET, and hence to the
+same end results.1
+The overspinning BTZ space-time is now obtained through identifications generated by a Killing
+field ξ, which in this case given by [3, 12]
+ξ = −a(J01 − J23) + b(J03 − J12),
+(10)
+which can be written as ξ =
+1
+2ωABJAB, where the antisymmetric matrix ωAB characterizes the
+identification. The Killing field in matrix form reads
+ξ =
+
+
+
+
+0
+−a
+0
+−b
+−a
+0
+−b
+0
+0
+b
+0
+−a
+b
+0
+−a
+0
+
+
+
+ .
+(11)
+1Without loss of generality, we will assume J > 0 for the rest of this work.
+3
+
+The identification in the embedding space R(2,2) under the action of the Killing field is a mapping
+defined by the matrix, H(ξ) = e2πξ, which takes the form
+H =
+
+
+
+
+C(a)c(b)
+−S(a)c(b)
+S(a)s(b)
+−C(a)s(b)
+−S(a)c(b)
+C(a)c(b)
+−C(a)s(b)
+S(a)s(b)
+−S(a)s(b)
+C(a)s(b)
+C(a)c(b)
+−S(a)c(b)
+C(a)s(b)
+−S(a)s(b)
+−S(a)c(b)
+C(a)c(b)
+
+
+
+ ,
+(12)
+where C(a) ≡ cosh(2πa), S(a) ≡ sinh(2πa) c(b) ≡ cos(2πb), and s(b) ≡ sin(2πb).
+An important feature of the Killing vector (10) is that the boost and rotation generators K ≡
+J01 − J23 and J ≡ J03 − J12 commute, [K, J] = 0. Consequently, H = e2πξ can be factored as
+H = Ha · Hb = Hb · Ha, where Ha = H|b=0 and Hb = H|a=0. Iterating the identification by H is
+equivalent to acting with
+Hn =
+
+
+
+
+C(na)c(nb)
+−S(na)c(nb)
+S(na)s(nb)
+−C(na)s(nb)
+−S(na)c(nb)
+C(na)c(nb)
+−C(na)s(nb)
+S(na)s(nb)
+−S(na)s(nb)
+C(na)s(nb)
+C(na)c(nb)
+−S(na)c(nb)
+C(na)s(nb)
+−S(na)s(nb)
+−S(na)c(nb)
+C(na)c(nb)
+
+
+
+ = Hn
+a · Hn
+b .
+(13)
+Quotienting a manifold by a rotation Killing vector requires the identification angle to be a
+rational fraction of 2π. Otherwise, each point is identified with infinitely many images which densely
+cover a circle, and the resulting image set would not be a smooth manifold [9]. This means that the
+coefficient b in (10) must be rational, namely,
+b = k/m,
+(14)
+with k, m relative primes. No restrictions are necessary for a, as boosts act transitively in a non-
+compact manner. Note that the m-th iteration produces a pure boost (and a rotation by 2kπ, which
+is equivalent to the identity, Hm
+b
+= 1). In fact, we can treat the rotated plane and the boosted
+plane separately by splitting the identification matrix as follows: consider writing n = qm+p, where
+p ∈ {0, 1, . . ., m − 1}, q ∈ {0, 1, . . ., ∞} and m is some positive integer.
+Hence, the powers of H = Ha · Hb can be arranged as follows
+1
+HaHb
+H2
+aH2
+b
+H3
+aH3
+b
+. . .
+Hm−1
+a
+Hm−1
+b
+Hm
+a
+Hm+1
+a
+Hb
+Hm+2
+a
+H2
+b
+Hm+3
+a
+H3
+b
+. . .
+H2m−1
+a
+Hm−1
+b
+H2m
+a
+H2m+1
+a
+Hb
+H2m+2
+a
+H2
+b
+H2m+3
+a
+H3
+b
+. . .
+H3m−1
+a
+Hm−1
+b
+...
+...
+...
+...
+...
+...
+.
+(15)
+Here each column corresponds to a fixed p and includes infinitely many boosts, while each row has
+a fixed q comprising a finite set of rotations. In this pattern, an interesting observation becomes
+apparent. First note that Ha is precisely the identification matrix of the rotating non-extremal BTZ
+black hole, and Hb the identification matrix of the rotating non-extremal naked singularity [9]. Now,
+using trigonometric identities, one can write in general, as can be seen in (15),
+Hqm+p = Hqm
+a
+Hp
+aHp
+b = Hq
+a·mHp
+aHp
+b ,
+(16)
+so that the p-th column reads
+Hp
+aHp
+b
+�
+1, H1
+a·m, H2
+a·m, H3
+a·m, · · ·
+�
+.
+(17)
+Or in other words, each column contains the powers of the identification matrix associated with the
+rotating non-extremal black hole, multiplied by some constant.
+4
+
+3
+Renormalized stress tensor
+To describe the quantum effects on the spacetime geometry, in particular the backreaction of the
+naked singularity to the presence of a quantum field, we consider the semi-classical Einstein equations
+Gµν − l−2gµν = κ ⟨Tµν⟩ ,
+(18)
+where ⟨Tµν⟩ is the renormalized expectation value of the quantum stress-energy tensor (RSET) of a
+conformally coupled scalar field [6, 7, 8, 9],
+κ ⟨Tµν(x)⟩ = πlP lim
+x′→x
+�
+3∇x
+µ∇x′
+ν − gµνgλρ∇x
+λ∇x′
+ρ − ∇x
+µ∇x
+ν − 1
+4l2 gµν
+�
+G(x, x′) , lP = ℏκ
+8π .
+(19)
+Using the method of images, the propagator, G(x, x′) = {φ(x), φ(x′)} is the anti-commutator of the
+scalar field, which takes the form [13, 14, 15, 16, 17, 9]
+G(x, x′) =
+1
+2
+√
+2π
+�
+n∈I
+Θ(σ(x, Hnx′))
+�
+σ(x, Hnx′)
+,
+(20)
+where σ(x, x′) is the chordal distance connecting x and x′, which can be expressed in terms of the
+corresponding embedding coordinates in R(2,2) as
+σ(x, x′) = 1
+2
+�
+−
+�
+X0 − X′0�2 +
+�
+X1 − X′1�2 +
+�
+X2 − X′2�2 −
+�
+X3 − X′3�2�
+.
+(21)
+The Heaviside step function Θ in (20) was introduced in [9] because σ(x, Hnx) can be negative in
+the rotating case. Calling dn(x) the cordal distance between a spacetime point and its nth image,
+dn = 2σ(x, Hnx) = 2l2 [−1 + cosh(2πan) cos(2πbn) − B(r) sinh(2πan) sin(2πbn)] ,
+(22)
+with
+B(r) = l2M − 2r2
+4abl2
+,
+(23)
+and the RSET takes the form [13, 9]
+κ ⟨Tµν⟩ = 3lP
+2
+�
+n∈I\{0}
+Θ(dn(x))
+�
+Sn
+µν − 1
+3gµνgλρSn
+λρ
+�
+,
+(24)
+with
+Sn
+ab = Hn
+ab
+d3/2
+n
++ 3Hn
+acXcH−n
+bd Xd − Hn
+acXcHn
+bdXd
+d5/2
+n
+.
+(25)
+The set I in the sum (24) includes all distinct images. With the splitting (16) between boosts (Ha)
+and rotations (Hb), one must sum over different ranges for q and p.
+3.1
+Explicit form for ⟨T µν⟩
+Note that for any rational value of b there are infinitely many values of n for which 2bn is an integer,
+which occurs for p = 0, which implies bn = kq and consequently the last term in (22) vanishes,
+making the distance function dn independent of r. This causes an infinite number of terms in the
+sum (24) to diverge, signaling a breakdown of the perturbative approach. This can be seen in the
+non-vanishing components of the stress-energy tensor,
+5
+
+κ ⟨T t
+t⟩ =lP l2
+8ab
+∞
+�
+n=1
+m∤n
+′ �
+6
+�
+a2 + b2�
+Bbn − 4ab¯bn + 12B¯an
+d5/2
+n
++
+�
+3
+�
+a2 − b2�
+B − 2ab
+�
+(¯cn − 8) +
+�
+3(a2 − b2) + 2abB
+�
+cnen
+d5/2
+n
+�
+,
+(26a)
+κ ⟨T t
+θ⟩ = − 3lP l3
+8ab
+∞
+�
+n=1
+m∤n
+′ 2
+��
+a2 − b2�
+B + 4ab
+�
+bn + 4Ban +
+�
+a2 + b2�
+[B (¯cn − 8) + encn]
+d5/2
+n
+,
+(26b)
+κ ⟨T r
+r⟩ =lP
+∞
+�
+n=1
+m∤n
+′ cn
+d3/2
+n
+(26c)
+κ ⟨T θ
+t⟩ =3lPl
+8ab
+∞
+�
+n=1
+m∤n
+′ 2
+��
+a2 − b2�
+B − 4ab
+�
+bn + 4Ban +
+�
+a2 + b2�
+[B (¯cn − 8) + cnen]
+d5/2
+n
+,
+(26d)
+κ ⟨T θ
+θ⟩ = − κ
+�
+⟨T t
+t⟩ + ⟨T r
+r⟩
+�
+,
+(26e)
+where
+′
+�
+n
+sn ≡ �
+n
+Θ(dn)sn, and
+an =a2 cos(4πbn) + b2 cosh(4πan) , ¯an = a2 cos(4πbn) − b2 cosh(4πan),
+(27a)
+bn = cos(4πbn) − cosh(4πan) ,
+¯bn = cos(4πbn) + cosh(4πan),
+(27b)
+cn =2 cosh(2πan) cos(2πbn) + 2 ,
+¯cn = 2 cosh(4πan) cos(4πbn) + 2,
+(27c)
+en =4 sinh(2πan) sin(2πbn).
+(27d)
+The presence of B(r) in the numerator of the ⟨T µ
+ν⟩ components makes them grow as r2 for large
+distance. Hence, as the denominators are independent of r for n = qm, these sums contain infinitely
+many asymptotically divergent terms. The problem is that to renormalize the stress-energy tensor
+using the Hadamard regularization scheme simply removes one divergent term corresponding to
+n = 0 (or p = q = 0) in the sum (20). However, we see that the stress energy tensor has infinitely
+many divergent terms, for p = 0 and all possible qs. A “natural” scheme to avoid the problem
+would be to eliminate the bs that generate the issue, but this would mean eliminating all rational bs,
+contradicting (14).
+It is still possible in principle that, in spite of the divergences in ⟨T µ
+ν⟩, they cancel out in the
+equations, yielding a finite result for the back reacted metric. We will see next that such cancellation
+does not occur, so that the field equations do not allow for a perturbative solution.
+3.2
+Backreacted metric
+The backreacted geometry is expected to belong in the same family of spherically symmetric sta-
+tionary BTZ metrics. It is therefore natural to assume the ansatz
+ds2 = − N(r)2f(r)dt2 + f(r)−1dr2 + r2 (dθ + k(r)dt)2 .
+(28)
+6
+
+Additionally, based on the previous results [9] we write
+N(r) =N0(r) + lP N1(r) + O(l2
+P ),
+(29)
+f(r) =f0(r) + lP f1(r) + O(l2
+P ),
+(30)
+k(r) =k0(r) + lP k1(r) + O(l2
+P ).
+(31)
+The zeroth order equations describe the unperturbed situation that yield the BTZ metric,
+N0(r) = 1,
+f0(r) = r2
+l2 − M + J2
+4r2 ,
+k0(r) = − J
+2r2 .
+(32)
+The first order corrections in lP of the field equations yield
+N1(r) = κ
+lP
+�
+dr
+r
+f0(r)
+�
+⟨T r
+r⟩ − ⟨T t
+t⟩ − J
+2r2 ⟨T t
+θ⟩
+�
++ K1,
+(33)
+f1(r) =
+�
+dr
+�
+−2f0(r)N ′
+1(r) +
+�J2
+r3 − 2M
+r
+�
+N1(r)
+(34)
++ 2
+r3
+�
+dr
+�
+2MrN1(r) + κ
+lP
+r3 ⟨T r
+r⟩
+��
++ K2
+r2 + K3,
+(35)
+Jk1(r) = − f1(r) − 2f0(r)N1(r) + 2
+�
+rdr
+� 2
+l2 N1(r) + κ
+lP
+⟨T r
+r⟩
+�
++ K4,
+(36)
+Here the integration constants must be chosen as Ki = 0 (i = 1, 2, 3, 4) so that the O(lP ) metric
+corrections vanish for ⟨T µ
+ν⟩ = 0. Even before integrating these expressions, it can be directly checked
+that the divergences of the stress-energy tensor do not cancel out, leading to unbounded results for
+N1, f1 and k1. Consequently, the perturbative ansatz (29 –31) does not work, since the first order
+corrections cannot be shown to be small.
+4
+Summary
+We have shown that a naked singularity of an overspinning BTZ geometry conformally coupled to
+a quantum scalar field does not lead to a renormalized stress-energy tensor. This causes incurable
+infinities to appear in the equations of motion and in the purportedly perturbative solutions. This
+is contrary to the previously studied cases of conical singularities, where the quantum corrections
+of the conformally coupled scalar field yields a finite renormalized stress-energy tensor and the
+resulting back-reacted geometry acquires a horizon, which provides a mechanism that enforces cosmic
+censorship [6, 7, 8, 9]. Our result indicates that the overspinning geometry is plagued by a more
+severe form of naked singularity, inaccessible by a perturbative approach. Consequently, it is not
+possible to claim that the singularity may become dressed by perturbative quantum corrections.
+Our result seems to indicate that coupling a conformal quantum scalar field to an overspinning
+geometry may cause the metric to be significantly different from the original BTZ metric. In any
+event, it is not possible to assert, as in the other cases of naked singularities, that quantum mechanics
+provides a cosmic censor in this case.
+It would be interesting to understand whether there is a more profound problem with this type
+of geometry, or if the strongly rotating behavior simply prevents the application of perturbative
+methods. Perhaps one way to approach this problem would be by numerical methods, hoping to
+get a better understanding of the nature of this particular type of singularity and to see if this is
+purely a problem of the perturbative approach, or if there is a more fundamental issue with the
+overspinning singularity.
+7
+
+Acknowledgements
+We thank C. Martínez, M. Hassaïne and Steen Ryom-Hansen for many enlightening discussions. OB
+is funded by the PhD scholarship of the University of Talca. This work has been partially funded
+by grant No 1220862 from ANID/Fondecyt.
+References
+[1]
+Roger Penrose. “Gravitational Collapse: the Role of General Relativity”. In: Nuovo Cimento
+Rivista Serie 1 (Jan. 1969), p. 252.
+[2]
+Máximo Bañados, Claudio Teitelboim, and Jorge Zanelli. “The Black hole in three-dimensional
+space-time”. In: Phys. Rev. Lett. 69 (1992), pp. 1849–1851. doi: 10.1103/PhysRevLett.69.1849.
+arXiv: hep-th/9204099.
+[3]
+Máximo Bañados et al. “Geometry of the (2+1) black hole”. In: Phys. Rev. D 48 (1993).
+[Erratum: Phys.Rev.D 88, 069902 (2013)], pp. 1506–1525. doi: 10.1103/PhysRevD.48.1506.
+arXiv: gr-qc/9302012.
+[4]
+Olivier Coussaert and Marc Henneaux. “Selfdual solutions of (2+1) Einstein gravity with a
+negative cosmological constant”. In: The Black Hole 25 Years After. Jan. 1994, pp. 25–39.
+arXiv: hep-th/9407181.
+[5]
+Eloy Ayon-Beato, Cristian Martinez, and Jorge Zanelli. “Birkhoff’s theorem for three-dimensional
+AdS gravity”. In: Phys. Rev. D 70 (2004), p. 044027. doi: 10.1103/PhysRevD.70.044027.
+arXiv: hep-th/0403227.
+[6]
+Marc Casals et al. “Quantum dress for a naked singularity”. In: Phys. Lett. B 760 (2016),
+pp. 244–248. doi: 10.1016/j.physletb.2016.06.044. arXiv: 1605.06078 [hep-th].
+[7]
+Marc Casals et al. “Quantum Backreaction on Three-Dimensional Black Holes and Naked Sin-
+gularities”. In: Phys. Rev. Lett. 118.13 (2017), p. 131102. doi: 10.1103/PhysRevLett.118.131102.
+arXiv: 1608.05366 [gr-qc].
+[8]
+Marc Casals et al. “Quantum fields as Cosmic Censors in (2 + 1)-dimensions”. In: Int. J. Mod.
+Phys. D 27.11 (2018). Ed. by L. C. B. Crispino et al., p. 1843011. doi: 10.1142/S0218271818430113.
+[9]
+Marc Casals et al. “Quantum-corrected rotating black holes and naked singularities in ( 2+1
+) dimensions”. In: Phys. Rev. D 99.10 (2019), p. 104023. doi: 10.1103/PhysRevD.99.104023.
+arXiv: 1902.01583 [hep-th].
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+Olivera Mišković and Jorge Zanelli. “On the negative spectrum of the 2+1 black hole”. In: Phys.
+Rev. D 79 (2009), p. 105011. doi: 10.1103/PhysRevD.79.10501.arXiv: arXiv:0904.0475[hep-th].
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+Roberto Emparan, Antonia Micol Frassino, and Benson Way. “Quantum BTZ black hole”. In:
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+[12]
+Matías Briceño, Cristián Martínez, and Jorge Zanelli. “Overspinning naked singularities in
+AdS3 spacetime”. In: (May 2021). arXiv: 2105.06488 [gr-qc].
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+Alan R. Steif. “The Quantum stress tensor in the three-dimensional black hole”. In: Phys. Rev.
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+In: Phys. Rev. D 18 (1978), p. 3565. doi: 10.1103/PhysRevD.18.3565.
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+10.1088/0264-9381/11/7/009. arXiv: 1901.00977 [gr-qc].
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+doi: 10.1103/PhysRevD.73.044027. arXiv: gr-qc/0511115.
+9
+

KdE3T4oBgHgl3EQfAQmY/content/tmp_files/load_file.txt

@@ -0,0 +1,360 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf,len=359
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='04256v1  [hep-th]  11 Jan 2023  Quantum backreaction for overspinning BTZ geometries  Olaf Baake2,1 ∗ and Jorge Zanelli1,3 †  1Centro de Estudios Científicos (CECs), Arturo Prat 514, Valdivia, Chile  2Instituto de Matemáticas, Universidad de Talca, Casilla 747, Talca 3460000, Chile  3Universidad San Sebastián, General Lagos 1163, Valdivia, Chile  January 12, 2023  Abstract  We examine the semiclassical backreaction of a conformally coupled scalar field on an over-  spinning BTZ geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' This extends the work done on a similar problem for (2 + 1)- AdS  geometries of the BTZ family with |M| > |J|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' The overspinning classical solutions corresponds  to |M| < |J| and possess a naked singularity at r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  Using the renormalized quantum  stress-energy tensor for a conformally coupled scalar field on such a spacetime, we obtain the  semiclassical Einstein equations, which we attempt to solve perturbatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' We show that the  stress-energy tensor is non-renormalizable in this approach, and consequently the perturbative  solution to the semiclassical equations in the overspinning case does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' This could be an  indication of the fact that the naked singularity at the center of an overspinning geometry is of  a more severe nature than the conical singularity found in the same family of BTZ geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  1  Introduction  Since the dawn of general relativity, many black hole solutions to Einstein’s field equations have been  found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' All these black holes contain a spacetime singularity hidden by an event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' However,  for some range of values of the integration constants (mass M, angular momentum J, electric charge  Q) these solutions have no event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' Although paradoxical, these naked singularities are exact  solutions to the classical equations of general relativity as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' In the vicinity of a naked singularity  causality and other physical laws can be arbitrarily violated, which is why Roger Penrose suggested  the existence of a (weak) cosmic censorship principle in nature [1], requiring singularities to be  hidden behind an event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' In that case, an outside observer would be causally disconnection  from the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  Classically, naked singularities cannot be ruled out on mathematical grounds, and it is difficult to  prove that every possible collapse process leads to the formation of an event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' The fact that  so far no naked singularities have been observed in the universe may be interpreted as an indication  that, in the strong gravity regime near a singularity, quantum gravity effects dominate eliminating  singularities altogether, or at least making sure that a horizon forms around them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  The accumulation of experiments and observations that confirm the predictions of general rel-  ativity puts very tight constraints on possible theories incorporating both general relativity and  ∗olaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='baake@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='com  †jorge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='zanelli@uss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='cl  1  quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' Since both theories are so well established in their regimes, it is sensible to look  for a common area where a semi-classical approach could be used to obtain a better understanding  of the issues at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' Calculating quantum effects on a curved background spacetime is notoriously  difficult, but in (2+1)-dimensional AdS spacetime this problem becomes significantly simpler and  still provide meaningful information to learn from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  The Bañados-Teitelboim-Zanelli (BTZ) black hole in (2+1)-dimensional AdS spacetime [2, 3],  obtained for M ≥ |J| are particularly interesting geometries in this respect, but these are not  the only solutions of physical interest in this theory and with the same global symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  Lo-  cally constant curvature 2+1 spacetimes include, besides the BTZ black hole family, the self-dual  Coussaert-Henneaux spacetimes [4], and the toroidal time-dependent geometries [5], with global  isometry groups SO(2) × R SO(2) × SO(2, 1) and SO(2) × SO(2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  Recently, the quantum back reaction on the classical singularities was studied for several geome-  tries, including static, rotating and extremal BTZ black holes, as well as for static and rotating  conical naked singularities [6, 7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' The naked singularities considered in these papers are contin-  uations of the BTZ spacetime to the case of negative mass [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' The interesting aspect of this result  is that the quantum fluctuations of a conformally coupled scalar field generate a non-vanishing stress  energy-momentum tensor that through Einstein’s equations produces aback-reacted geometry with  a horizon of order Planck length in radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' This dressing up of the naked singularity, turning it into  a black hole, could be viewed as a mechanism that implements cosmic censorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' These results  have also been confirmed by an alternative holographic approach in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  Here we are concerned with the overspinning BTZ spacetime, which occurs if the absolute value  of the angular momentum is greater than that of the mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' This geometry is also endowed with a  naked singularity at r = 0, as in the case of the conical singularity obtained for M ≤ −|J|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  We show that the stress-energy tensor contains incurable divergences, making the perturbative  ansatz to the semiclassical equations of motion ill-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' While the equations of motion can still be  formally integrated, the first order corrections to the metric functions would become large, further  demonstrating the inapplicability of a perturbative approach to this type of geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' This strongly  suggests that the naked singularity of an overspinning geometry is of a more severe nature than  the conical singularities appearing in the other BTZ geometries so that they cannot be cured by a  perturbative quantum censor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  2  Overspinning BTZ space-time  The rotating BTZ metric [2, 3], is given by  ds2 = −  �r2  l2 − M  �  dt2 − Jdtdθ +  �r2  l2 − M + J2  4r2  �−1  dr2 + r2dθ2,  (1)  where the coordinate ranges are: −∞ < t < ∞, 0 < r < ∞ and 0 ≤ θ < 2π, Λ = −l−2 is the  cosmological constant, and M and J are mass and angular momentum respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' This metric  describes different spacetimes that can be classified by the values of M and J which determine the  nature of the four roots of the equation grr = 0,  λ± = l  2  ��  M + J  l ±  �  M − J  l  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (2)  These roots are real for M ≥ |J|/l (black holes) and take complex values for M < |J|/l (naked  singularities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' The full classification is explained in detail in [3], but here we will consider the so-  called overspinning geometry (|M|l < |J|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' This geometry was examined in [12] through the study  2  of classical geodesics around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' In particular, we will analyze the back reaction of the geometry to  the presence of a conformally coupled quantum scalar field, following the steps in [6, 7, 8, 9], where  the back reaction for conical naked singularities in the parameter range M ≤ −|J| was studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  The starting point of the analysis is the observation that the BTZ spacetimes (1) are quotients of  the universal covering of anti-de Sitter space-time (CAdS3) by an appropriate Killing vector field [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  The constant negative curvature spacetime AdS3 is defined by a pseudosphere of radius l embedded  in R(2,2) as  ηABXAXB = −  �  X0�2 +  �  X1�2 +  �  X2�2 −  �  X3�2 = −l2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (3)  The metric reads  ηABdXAdXB = −  �  dX0�2 +  �  dX1�2 +  �  dX2�2 −  �  dX3�2 ,  (4)  where the embedding coordinates XA must be specified as functions of (t, r, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' As shown in [12],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  the overspinning geometry (1) with |M| < |J| corresponds to embedding coordinates given by  X0 = l  2  √  A + 1 cosh [a (t/l − θ)] {cos [b (θ + t/l)] − sin [b (θ + t/l)]}  +ǫ l  2  √  A − 1 sinh [a (t/l − θ)] {sin [b (θ + t/l)] + cos [b (θ + t/l)]} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (5)  X1 = l  2  √  A + 1 sinh [a (t/l − θ)] {cos [b (θ + t/l)] − sin [b (θ + t/l)]}  +ǫ l  2  √  A − 1 cosh [a (t/l − θ)] {sin [b (θ + t/l)] + cos [b (θ + t/l)]} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (6)  X2 = l  2  √  A + 1 sinh [a (t/l − θ)] {sin [b (θ + t/l)] + cos [b (θ + t/l)]}  −ǫ l  2  √  A − 1 cosh [a (t/l − θ)] {cos [b (θ + t/l)] − sin [b (θ + t/l)]} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (7)  X3 = l  2  √  A + 1 cosh [a (t/l − θ)] {sin [b (θ + t/l)] + cos [b (θ + t/l)]}  −ǫ l  2  √  A − 1 sinh [a (t/l − θ)] {cos [b (θ + t/l)] − sin [b (θ + t/l)]} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (8)  where  a =  �  |J|/l + M  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  b =  �  |J|/l − M  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  A =  2  �  J2  4 + r4  l2 − Mr2  √  J2 − l2M 2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (9)  with ǫ = sign(M − r2/l2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' Note that both cases (ǫ = ±1) lead to the same RSET, and hence to the  same end results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='1  The overspinning BTZ space-time is now obtained through identifications generated by a Killing  field ξ, which in this case given by [3, 12]  ξ = −a(J01 − J23) + b(J03 − J12),  (10)  which can be written as ξ =  1  2ωABJAB, where the antisymmetric matrix ωAB characterizes the  identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' The Killing field in matrix form reads  ξ =  \uf8eb  \uf8ec  \uf8ec  \uf8ed  0  −a  0  −b  −a  0  −b  0  0  b  0  −a  b  0  −a  0  \uf8f6  \uf8f7  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (11)  1Without loss of generality, we will assume J > 0 for the rest of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  3  The identification in the embedding space R(2,2) under the action of the Killing field is a mapping  defined by the matrix, H(ξ) = e2πξ, which takes the form  H =  \uf8eb  \uf8ec  \uf8ec  \uf8ed  C(a)c(b)  −S(a)c(b)  S(a)s(b)  −C(a)s(b)  −S(a)c(b)  C(a)c(b)  −C(a)s(b)  S(a)s(b)  −S(a)s(b)  C(a)s(b)  C(a)c(b)  −S(a)c(b)  C(a)s(b)  −S(a)s(b)  −S(a)c(b)  C(a)c(b)  \uf8f6  \uf8f7  \uf8f7  \uf8f8 ,  (12)  where C(a) ≡ cosh(2πa), S(a) ≡ sinh(2πa) c(b) ≡ cos(2πb), and s(b) ≡ sin(2πb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  An important feature of the Killing vector (10) is that the boost and rotation generators K ≡  J01 − J23 and J ≡ J03 − J12 commute, [K, J] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' Consequently, H = e2πξ can be factored as  H = Ha · Hb = Hb · Ha, where Ha = H|b=0 and Hb = H|a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' Iterating the identification by H is  equivalent to acting with  Hn =  \uf8eb  \uf8ec  \uf8ec  \uf8ed  C(na)c(nb)  −S(na)c(nb)  S(na)s(nb)  −C(na)s(nb)  −S(na)c(nb)  C(na)c(nb)  −C(na)s(nb)  S(na)s(nb)  −S(na)s(nb)  C(na)s(nb)  C(na)c(nb)  −S(na)c(nb)  C(na)s(nb)  −S(na)s(nb)  −S(na)c(nb)  C(na)c(nb)  \uf8f6  \uf8f7  \uf8f7  \uf8f8 = Hn  a · Hn  b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (13)  Quotienting a manifold by a rotation Killing vector requires the identification angle to be a  rational fraction of 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' Otherwise, each point is identified with infinitely many images which densely  cover a circle, and the resulting image set would not be a smooth manifold [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' This means that the  coefficient b in (10) must be rational, namely,  b = k/m,  (14)  with k, m relative primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' No restrictions are necessary for a, as boosts act transitively in a non-  compact manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' Note that the m-th iteration produces a pure boost (and a rotation by 2kπ, which  is equivalent to the identity, Hm  b  = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' In fact, we can treat the rotated plane and the boosted  plane separately by splitting the identification matrix as follows: consider writing n = qm+p, where  p ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=', m − 1}, q ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=', ∞} and m is some positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  Hence, the powers of H = Ha · Hb can be arranged as follows  1  HaHb  H2  aH2  b  H3  aH3  b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  Hm−1  a  Hm−1  b  Hm  a  Hm+1  a  Hb  Hm+2  a  H2  b  Hm+3  a  H3  b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  H2m−1  a  Hm−1  b  H2m  a  H2m+1  a  Hb  H2m+2  a  H2  b  H2m+3  a  H3  b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  H3m−1  a  Hm−1  b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (15)  Here each column corresponds to a fixed p and includes infinitely many boosts, while each row has  a fixed q comprising a finite set of rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' In this pattern, an interesting observation becomes  apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' First note that Ha is precisely the identification matrix of the rotating non-extremal BTZ  black hole, and Hb the identification matrix of the rotating non-extremal naked singularity [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' Now,  using trigonometric identities, one can write in general, as can be seen in (15),  Hqm+p = Hqm  a  Hp  aHp  b = Hq  a·mHp  aHp  b ,  (16)  so that the p-th column reads  Hp  aHp  b  �  1, H1  a·m, H2  a·m, H3  a·m, · · ·  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (17)  Or in other words, each column contains the powers of the identification matrix associated with the  rotating non-extremal black hole, multiplied by some constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  4  3  Renormalized stress tensor  To describe the quantum effects on the spacetime geometry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' in particular the backreaction of the  naked singularity to the presence of a quantum field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' we consider the semi-classical Einstein equations  Gµν − l−2gµν = κ ⟨Tµν⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (18)  where ⟨Tµν⟩ is the renormalized expectation value of the quantum stress-energy tensor (RSET) of a  conformally coupled scalar field [6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' 9],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  κ ⟨Tµν(x)⟩ = πlP lim  x′→x  �  3∇x  µ∇x′  ν − gµνgλρ∇x  λ∇x′  ρ − ∇x  µ∇x  ν − 1  4l2 gµν  �  G(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' x′) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' lP = ℏκ  8π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (19)  Using the method of images, the propagator, G(x, x′) = {φ(x), φ(x′)} is the anti-commutator of the  scalar field, which takes the form [13, 14, 15, 16, 17, 9]  G(x, x′) =  1  2  √  2π  �  n∈I  Θ(σ(x, Hnx′))  �  σ(x, Hnx′)  ,  (20)  where σ(x, x′) is the chordal distance connecting x and x′, which can be expressed in terms of the  corresponding embedding coordinates in R(2,2) as  σ(x, x′) = 1  2  �  −  �  X0 − X′0�2 +  �  X1 − X′1�2 +  �  X2 − X′2�2 −  �  X3 − X′3�2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (21)  The Heaviside step function Θ in (20) was introduced in [9] because σ(x, Hnx) can be negative in  the rotating case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' Calling dn(x) the cordal distance between a spacetime point and its nth image,  dn = 2σ(x, Hnx) = 2l2 [−1 + cosh(2πan) cos(2πbn) − B(r) sinh(2πan) sin(2πbn)] ,  (22)  with  B(r) = l2M − 2r2  4abl2  ,  (23)  and the RSET takes the form [13, 9]  κ ⟨Tµν⟩ = 3lP  2  �  n∈I\\{0}  Θ(dn(x))  �  Sn  µν − 1  3gµνgλρSn  λρ  �  ,  (24)  with  Sn  ab = Hn  ab  d3/2  n  + 3Hn  acXcH−n  bd Xd − Hn  acXcHn  bdXd  d5/2  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (25)  The set I in the sum (24) includes all distinct images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' With the splitting (16) between boosts (Ha)  and rotations (Hb), one must sum over different ranges for q and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='1  Explicit form for ⟨T µν⟩  Note that for any rational value of b there are infinitely many values of n for which 2bn is an integer,  which occurs for p = 0, which implies bn = kq and consequently the last term in (22) vanishes,  making the distance function dn independent of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' This causes an infinite number of terms in the  sum (24) to diverge, signaling a breakdown of the perturbative approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' This can be seen in the  non-vanishing components of the stress-energy tensor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  5  κ ⟨T t  t⟩ =lP l2  8ab  ∞  �  n=1  m∤n  ′ �  6  �  a2 + b2�  Bbn − 4ab¯bn + 12B¯an  d5/2  n  +  �  3  �  a2 − b2�  B − 2ab  �  (¯cn − 8) +  �  3(a2 − b2) + 2abB  �  cnen  d5/2  n  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (26a)  κ ⟨T t  θ⟩ = − 3lP l3  8ab  ∞  �  n=1  m∤n  ′ 2  ��  a2 − b2�  B + 4ab  �  bn + 4Ban +  �  a2 + b2�  [B (¯cn − 8) + encn]  d5/2  n  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (26b)  κ ⟨T r  r⟩ =lP  ∞  �  n=1  m∤n  ′ cn  d3/2  n  (26c)  κ ⟨T θ  t⟩ =3lPl  8ab  ∞  �  n=1  m∤n  ′ 2  ��  a2 − b2�  B − 4ab  �  bn + 4Ban +  �  a2 + b2�  [B (¯cn − 8) + cnen]  d5/2  n  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (26d)  κ ⟨T θ  θ⟩ = − κ  �  ⟨T t  t⟩ + ⟨T r  r⟩  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (26e)  where  ′  �  n  sn ≡ �  n  Θ(dn)sn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' and  an =a2 cos(4πbn) + b2 cosh(4πan) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' ¯an = a2 cos(4πbn) − b2 cosh(4πan),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (27a)  bn = cos(4πbn) − cosh(4πan) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  ¯bn = cos(4πbn) + cosh(4πan),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (27b)  cn =2 cosh(2πan) cos(2πbn) + 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  ¯cn = 2 cosh(4πan) cos(4πbn) + 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (27c)  en =4 sinh(2πan) sin(2πbn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (27d)  The presence of B(r) in the numerator of the ⟨T µ  ν⟩ components makes them grow as r2 for large  distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' Hence, as the denominators are independent of r for n = qm, these sums contain infinitely  many asymptotically divergent terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' The problem is that to renormalize the stress-energy tensor  using the Hadamard regularization scheme simply removes one divergent term corresponding to  n = 0 (or p = q = 0) in the sum (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' However, we see that the stress energy tensor has infinitely  many divergent terms, for p = 0 and all possible qs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' A “natural” scheme to avoid the problem  would be to eliminate the bs that generate the issue, but this would mean eliminating all rational bs,  contradicting (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  It is still possible in principle that, in spite of the divergences in ⟨T µ  ν⟩, they cancel out in the  equations, yielding a finite result for the back reacted metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' We will see next that such cancellation  does not occur, so that the field equations do not allow for a perturbative solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='2  Backreacted metric  The backreacted geometry is expected to belong in the same family of spherically symmetric sta-  tionary BTZ metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' It is therefore natural to assume the ansatz  ds2 = − N(r)2f(r)dt2 + f(r)−1dr2 + r2 (dθ + k(r)dt)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (28)  6  Additionally, based on the previous results [9] we write  N(r) =N0(r) + lP N1(r) + O(l2  P ),  (29)  f(r) =f0(r) + lP f1(r) + O(l2  P ),  (30)  k(r) =k0(r) + lP k1(r) + O(l2  P ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (31)  The zeroth order equations describe the unperturbed situation that yield the BTZ metric,  N0(r) = 1,  f0(r) = r2  l2 − M + J2  4r2 ,  k0(r) = − J  2r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (32)  The first order corrections in lP of the field equations yield  N1(r) = κ  lP  �  dr  r  f0(r)  �  ⟨T r  r⟩ − ⟨T t  t⟩ − J  2r2 ⟨T t  θ⟩  �  + K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (33)  f1(r) =  �  dr  �  −2f0(r)N ′  1(r) +  �J2  r3 − 2M  r  �  N1(r)  (34)  + 2  r3  �  dr  �  2MrN1(r) + κ  lP  r3 ⟨T r  r⟩  ��  + K2  r2 + K3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (35)  Jk1(r) = − f1(r) − 2f0(r)N1(r) + 2  �  rdr  � 2  l2 N1(r) + κ  lP  ⟨T r  r⟩  �  + K4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  (36)  Here the integration constants must be chosen as Ki = 0 (i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' 4) so that the O(lP ) metric  corrections vanish for ⟨T µ  ν⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' Even before integrating these expressions, it can be directly checked  that the divergences of the stress-energy tensor do not cancel out, leading to unbounded results for  N1, f1 and k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' Consequently, the perturbative ansatz (29 –31) does not work, since the first order  corrections cannot be shown to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  4  Summary  We have shown that a naked singularity of an overspinning BTZ geometry conformally coupled to  a quantum scalar field does not lead to a renormalized stress-energy tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' This causes incurable  infinities to appear in the equations of motion and in the purportedly perturbative solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' This  is contrary to the previously studied cases of conical singularities, where the quantum corrections  of the conformally coupled scalar field yields a finite renormalized stress-energy tensor and the  resulting back-reacted geometry acquires a horizon, which provides a mechanism that enforces cosmic  censorship [6, 7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' Our result indicates that the overspinning geometry is plagued by a more  severe form of naked singularity, inaccessible by a perturbative approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' Consequently, it is not  possible to claim that the singularity may become dressed by perturbative quantum corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  Our result seems to indicate that coupling a conformal quantum scalar field to an overspinning  geometry may cause the metric to be significantly different from the original BTZ metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' In any  event, it is not possible to assert, as in the other cases of naked singularities, that quantum mechanics  provides a cosmic censor in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  It would be interesting to understand whether there is a more profound problem with this type  of geometry, or if the strongly rotating behavior simply prevents the application of perturbative  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' Perhaps one way to approach this problem would be by numerical methods, hoping to  get a better understanding of the nature of this particular type of singularity and to see if this is  purely a problem of the perturbative approach, or if there is a more fundamental issue with the  overspinning singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  7  Acknowledgements  We thank C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' Martínez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' Hassaïne and Steen Ryom-Hansen for many enlightening discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' OB  is funded by the PhD scholarship of the University of Talca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' This work has been partially funded  by grant No 1220862 from ANID/Fondecyt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  References  [1]  Roger Penrose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' “Gravitational Collapse: the Role of General Relativity”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' In: Nuovo Cimento  Rivista Serie 1 (Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
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+page_content='  [2]  Máximo Bañados, Claudio Teitelboim, and Jorge Zanelli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' “The Black hole in three-dimensional  space-time”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
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+page_content=' 1849–1851.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
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+page_content='  arXiv: hep-th/9204099.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='  [3]  Máximo Bañados et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' “Geometry of the (2+1) black hole”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
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+page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
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+page_content=' 1506–1525.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content='48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
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+page_content='  arXiv: gr-qc/9302012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
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+page_content=' “Vacuum polarization near asymptotically anti-de Sit-  ter black holes in odd dimensions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
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+page_content=' arXiv: 1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
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+page_content=' “Off-diagonal coefficients of the Dewitt-Schwinger and  Hadamard representations of the Feynman propagator”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
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+page_content='  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
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+page_content='044027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
+page_content=' arXiv: gr-qc/0511115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'}
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L9E4T4oBgHgl3EQfiw1J/content/tmp_files/2301.05136v1.pdf.txt

@@ -0,0 +1,675 @@
+Received ;
+Revised ;
+Accepted
+DOI: xxx/xxxx
+PROCEEDINGS IWARA 2022
+Light vector meson photoproduction in ultraperipheral heavy ion
+collisions at the LHC within the Reggeometric Pomeron approach
+László Jenkovszky1 | Érison S. Rocha2 | Magno V. T. Machado*2
+1Bogolyubov ITP, National Academy of
+Sciences of Ukraine, Kiev 03143, Ukraine
+2HEP Phenomenology Group, Instituto de
+Física UFRGS, RS, Brazil
+Correspondence
+*M.V.T. Machado. Email:
+[email protected]
+Funding Information
+National Academy of Science of
+Ukraine, 1230/22-1 Fundamental
+Properties of Matter. Coordination
+for the Improvement of Higher Edu-
+cation Personnel (CAPES/Brazil),
+Finance Code 001. National Coun-
+cil for Scientific and Technolog-
+ical Development (CNPq/Brazil),
+306101/2018-1.
+By using the Reggeometric Pomeron model for vector meson production which
+successfully describes the high energy lepton-nucleon data, we analyse the light
+meson production in ultra-peripheral heavy ion collisions at the Large Hadron Col-
+lider (LHC). The rapidity distributions for 휌 and 휙 photoproduction in lead-lead,
+xenon-xenon and oxygen-oxygen collisions are investigated.
+KEYWORDS:
+ultra-peripheral heavy ion collisions, vector meson photoproduction, Regge phenomenology, Large
+Hadron Collider
+1
+INTRODUCTION
+The exclusive light vector meson (푉 ) photoproduction has
+been studied in recent years both experimentally and theoreti-
+cally (Acharya et al., 2020, 2021; Andreev et al., 2020; S. Klein
+et al., 2020; Sirunyan et al., 2019). The process has not asso-
+ciated hard perturbative Quantum Chromodynamics (pQCD)
+scale in the photoproduction limit, 푄2 → 0. Here, 푄2 is the so
+called photon virtuality in the process 훾∗+푁 → 푉 +푁. There-
+fore, light mesons can be used to test the non-perturbative
+regime of the strong interactions. Within the parton satura-
+tion formalism the transition between the region described by
+pQCD and the non-perturbative regime is interpreted in terms
+of the nucleon QCD saturation scale (Morreale & Salazar,
+2021), 푄푠(푥) ∼ 푥−0.3, with 푥 being the invariant Bjorken
+kinematic variable. In the vector meson electroproduction off
+nucleon target, 푥 = (푀2
+푉 + 푄2)∕(푊 2
+훾푁 + 푄2), where 푊훾푁 is
+the centre-of-mass energy of the photon-nucleon system. In the
+QCD color dipole picture (Chen & Mueller, 1995; Nemchik,
+Nikolaev, Predazzi, & Zakharov, 1996; Nikolaev & Zakharov,
+1994) 푄푠 characterizes the boundary on the maximum phase-
+space gluon density to be reached in the wave-function of the
+nucleon. In this framework, the light meson photoproduction
+dynamics at the present accelerator energies can be treated per-
+turbatively as 푄푠 reaches values ≲ 1 GeV at very high energies.
+The perturbative description is even improved in case of scat-
+tering off nuclei of atomic number 퐴 as the nuclear saturation
+scale, 푄2
+푠,퐴 ∝ 퐴1∕3푄2
+푠, is enhanced regarding the proton. On
+the other hand, it is well known that this approach has been
+unable to describe precisely the total photoproduction cross
+section and 휌 production at 푄2 = 0 GeV2 (Forshaw, Kerley,
+& Shaw, 1999; Gonçalves & Moreira, 2020). In fact, non-
+perturbative corrections are necessary and they are embedded
+in the photon wave-function, 휓훾
+푇 , for color dipoles containing
+large transverse size.
+In the soft physics sector, the Regge phenomenology
+(Jenkovszky, Schicker, & Szanyi, 2018) is a well founded and
+appropriated formalism to describe exclusive diffractive pro-
+cesses, including the light meson photoproduction. The vector
+meson production amplitude is written in a Regge-factorized
+arXiv:2301.05136v1  [hep-ph]  12 Jan 2023
+
+2
+structure with the corresponding coupling of particles to the
+Pomeron. The introduction of a perturbative scale depen-
+dence suitable for electroproduction can be constructed based
+on geometric arguments. The Reggeometric Pomeron (RP)
+model (Fazio, Fiore, Jenkovszky, & Salii, 2014; Fazio, Fiore,
+Lavorini, Jenkovszky, & Salii, 2013) is one example of such
+a class of phenomenological models. The RP model does a
+good job in describing both photo and electroproduction at
+the DESY-HERA energy regime considering a nucleon target.
+The possibility for testing these models in the coherent vec-
+tor meson production in ultraperipheral heavy ion collisions
+(UPCs) is a reality nowadays. The basic argument is that the
+production cross section in nucleus-nucleus (퐴퐴) collisions
+can be factorized in terms of the equivalent flux of photons of
+the colliding nucleus and the photon-target production cross
+section (S. Klein & Steinberg, 2020).
+In this contribution, the light meson photoproduction in
+nucleus-nucleus UPCs collisions is investigated. The focus is
+on the energies and nuclear species in the heavy ion collisions
+at the Large Hadron Collider (LHC). The theoretical input is
+the description based on the Reggeometric Pomeron model for
+the elastic differential and integrated total cross section in the
+(quasi-real) photon interaction with nucleons. The parameters
+of the model are consistent with the measurements performed
+by HERA-H1 (Andreev et al., 2020) and CMS collabora-
+tions (Sirunyan et al., 2019) as shown in Ref. (Jenkovszky,
+Rocha, & Machado, 2022). The nuclear coherent cross section
+is then obtained by using Vector Dominance Model (VDM)
+and Glauber multiple scattering theory. Predictions are per-
+formed for 휌 and 휙 production in 퐴퐴 UPCs at the LHC. In
+particular, results are compared to the measurements in PbPb
+and XeXe UPCs done by ALICE Collaboration (Acharya et
+al., 2020, 2021) for the energies of √푠NN = 5.02 TeV and
+√푠NN = 5.44 TeV, respectively. Theoretical estimates for the
+cross section in OO collisions are also presented. The study is
+based on earlier works (Jenkovszky, Libov, & Machado, 2022a,
+2022b; Jenkovszky, Rocha, & Machado, 2022) by the authors.
+The work has been organized as follows. In Sec. 2 we shortly
+review the exclusive vector meson production, 훾+푝 → 푉 +푝, in
+the context of the Reggeometric Pomeron model. Afterwards,
+using VDM model and Glauber formalism for nuclear shad-
+owing, the expression for the coherent nuclear cross section is
+obtained. In section 3 the calculations are compared to avail-
+able experimental measurements in PbPb and XeXe UPCs
+collisions at the LHC. Prediction are done for future light ion
+runs like oxygen-oxygen collisions. Furthermore, discussion
+on the theoretical uncertainties is presented. In section 4 the
+key results are summarized.
+2
+THEORETICAL FRAMEWORK
+Exclusive vector meson photoproduction process, 훾 + 푝 →
+푉 + 푝, will be described by using a model based on Regge
+phenomenology, namely the Reggeometric Pomeron model. It
+is also able to describe electroproduction data as discussed in
+what follows. In general case, the hardness scale is given by
+̃푄2 = 푄2 + 푀2
+푉 .
+The elastic differential cross section, 푑휎푒푙∕푑푡, related to the
+single-component Reggeometric model in a given scale ̃푄2 is
+given by (Fazio et al., 2014, 2013):
+푑휎푒푙
+푑푡
+=
+퐴2
+0 exp
+[
+퐵0(̃
+푄2) 푡
+]
+(
+1 +
+̃
+푄2
+푄2
+0
+)2푛
+(푊 2
+훾푝
+푊 2
+0
+)2(훼(푡)−1)
+,
+(1)
+퐵0(̃
+푄2) = 4
+(
+푎
+̃
+푄2 +
+푏
+2푚2
+푁
+)
+,
+(2)
+where the quantity 퐵0(̃
+푄2) reflects the geometrical nature of
+the model and 훼(푡) denotes the effective Pomeron (퐼푃) trajec-
+tory. The first and second term in Eq. (2) correspond to the
+effective sizes of the 훾퐼푃 푉 and 푝퐼푃 푝 vertices, respectively. In
+the formula above, 푊0 = 1 GeV and 푚푁 is the nucleon mass.
+It is assumed a linear Pomeron trajectory, 훼(푡) = 훼0 +훼′푡, with
+an effective Pomeron intercept 훼0.
+Accordingly, the integrated cross section is written as,
+휎(훾∗ + 푝 → 푉 + 푝) =
+퐴2
+0
+(
+1 +
+̃푄2
+̃푄2
+0
+)2푛
+(푊훾푝∕푊0
+)4(훼0−1)
+퐵
+(
+푊훾푝, ̃푄2
+)
+, (3)
+퐵
+(
+푊훾푝, ̃푄2)
+= 퐵0(̃
+푄2) + 4훼′ ln
+(푊훾푝
+푊0
+)
+.
+(4)
+In the photoproduction limit one has ̃푄2 = 푀2
+푉 and the param-
+eters of the model for 휌 and 휙 production are presented in Table
+1 . They have been determined (Fazio et al., 2014) by using
+DESY-HERA measurements (Aaron et al., 2010; Aid et al.,
+1996; Breitweg et al., 1998, 2000; Derrick et al., 1995).
+Now the expressions for the nuclear coherent cross sections
+are presented. Following the STARLIGHT Monte Carlo gen-
+erator approach for UPCs processes (S. R. Klein, Nystrand,
+Seger, Gorbunov, & Butterworth, 2017), nuclear effects for the
+process, 훾 + 퐴 → 푉 + 퐴 are described here by vector domi-
+nance model (Bauer, Spital, Yennie, & Pipkin, 1978) and the
+classical mechanics Glauber formula for multiple scattering of
+the vector meson in the nuclear medium. At 푡 = 0 the differen-
+tial cross section is obtained by using the Optical theorem for
+scattering in a nucleus and VDM as follows,
+d휎 (훾 + 퐴 → 푉 + 퐴)
+d푡
+||||푡=0
+= 훼푒푚
+4푓 2
+푉
+휎2
+푡표푡 (푉 퐴) ,
+(5)
+휎푡표푡 (푉 퐴) = ∫ d2b [1 − 푒−휎푡표푡(푉 푝)푇퐴(b)] ,(6)
+
+3
+TABLE 1 Values of the parameters for the Reggeometric Pomeron model (Fazio et al., 2014).
+Meson
+퐴0
+[ √
+nb
+GeV
+]
+̃
+푄2
+0
+[GeV2]
+푛
+훼0
+훼′ [GeV−2]
+푎
+푏
+휌
+344 ± 376
+0.29 ± 0.14
+1.24 ± 0.07
+1.16 ± 0.14
+0.21 ± 0.53
+0.60 ± 0.33
+0.9 ± 4.3
+휙
+58 ± 112
+0.89 ± 1.40
+1.30± 0.28
+1.14± 0.19
+0.17 ± 0.78
+0.0 ± 19.8
+1.34± 5.09
+where 푇퐴(푏) is the nuclear thickness function and 푓푉 is
+the vector-meson coupling. The values 푓 2
+휌 ∕4휋 = 2.02 and
+푓 2
+휙∕4휋 = 13.7 are considered in calculations, respectively. For
+light mesons, 휎푡표푡(푉 푝) is large and the cross section 휎푡표푡(푉 퐴)
+is approximately the geometric cross section. It is also almost
+energy independent (Jenkovszky, Rocha, & Machado, 2022).
+The input for the Glauber model calculation in Eq. (6) is the
+effective vector meson–nucleon cross for the process 푉 + 푝 →
+푉 + 푝, which is given by:
+휎푡표푡 (푉 푝) =
+√
+4푓 2
+푉
+훼푒푚
+d휎 (훾 + 푝 → 푉 + 푝)
+d푡
+||||푡=0
+,
+(7)
+where the differential cross section coming from the Reggeo-
+metric Pomeron model, Eq. (1), will be introduced in Eq. (7) .
+The corresponding integrated cross section is given by:
+휎(훾 + 퐴 → 푉 + 퐴) = 푑휎(훾 + 퐴 → 푉 + 퐴)
+푑푡
+||||푡=0
+×
+∞
+∫
+푡푚푖푛
+d|푡| ||퐹퐴 (푡)||
+2 ,
+(8)
+where the quantity 퐹퐴 is the nuclear form factor. It is taken
+into account an analytic form factor given by a hard sphere
+of radius, 푅퐴 = 푟0퐴1∕3 fm (푟0 ≃ 1.2 fm), convoluted with a
+Yukawa potential with range 푎 (Davies & Nix, 1976),
+퐹퐴(|푞|) = 4휋휌0
+퐴|푞3|
+(
+1
+1 + 푎2푞2
+)
+× [sin (|푞|푅퐴) − |푞|푅퐴 cos (|푞|푅퐴)] ,
+(9)
+where 푞 is the momentum transfer, 휌0 = 3퐴∕(4휋푅3
+퐴) fm−3 and
+푎 = 0.7 fm.
+In the calculations the reggeon contribution is added to
+the photoproduction off nucleons. The corresponding cross
+section is parameterized as,
+푑휎퐼푅(훾푝 → 푉 푃)
+푑푡
+|||||푡=0
+= 푏푉 푌 푊 −휂
+훾푝 ,
+(10)
+where the constants 푏푉
+= 11 GeV−2, 푌
+= 26.0 휇b and
+휂 = 1.23 have been considered for the 휌 production. For the
+휙, meson exchange is strongly suppressed, and the reaction
+occurs only through 퐼푃-exchange.
+The 퐴-dependence of the cross section for the coherent pro-
+duction of 휌 meson, 휎(훾 + 퐴 → 휌 + 퐴), is presented in Fig.
+FIGURE 1 The 퐴-dependence of the cross section for the
+coherent production of 휌 meson from Reggeometric Pomeron
+model at the LHC. Data from ALICE Collaboration (Acharya
+et al., 2021).
+1 . Comparison is done with the extracted values of the coher-
+ent cross section performed in Ref. (Acharya et al., 2021) by
+ALICE Collaboration using the measured data on UPCs at the
+LHC (PbPb collisions at 5.02 TeV and XeXe at 5.44 TeV).
+The description is quite reasonable for the nuclear dependence.
+At central rapidity, 푦 = 0, the photon-nucleon centre-of-mass
+energy squared is 푊 2
+훾푁 = 푀푉
+√푠NN. For xenon the cross
+section corresponds to 푊훾푁 = 65 GeV and 휎(훾Xe → 휌Xe) ≃
+1.12 ± 0.21 mb. The predicted values from the Reggeomet-
+ric Pomeron model is 1.07 mb. For lead, the data is 휎(훾Pb →
+휌Pb) ≃ 2.09 ± 0.16 mb for energy 푊훾푁 = 62 GeV and the
+prediction 1.54 mb. Theoretical calculation underestimates the
+extracted 훾Pb cross section, which suggests a strong nuclear
+shadowing correction for very large nucleus in the formalism
+considered for the study.
+
+3,0
+ALICE-Xe
+2,5
+ALICE-Pb
+ReggeometricPomeron model
+[mb]
++A)
+P1,5
+个
+V+i)
+1,0
+a
+0.5
+0,0
+1
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+220
+240
+A4
+FIGURE 2 Rapidity distributions for the exclusive 휌 meson photoproduction in ultraperipheral PbPb (left panel), XeXe (central
+panel and OO (right panel) collisions considering the Reggeometric Pomeron model. Prediction are done for the current run
+(solid lines) and future HL-LHC run (dashed lines). Comparison is done to ALICE Collaboration data (Acharya et al., 2020,
+2021).
+3
+RESULTS AND DISCUSSIONS
+The rapidity distribution for meson production in nucleus-
+nucleus UPCs takes a factorized form in the Equivalent Photon
+Approximation (EPA). The expression is given by:
+푑휎(퐴 + 퐴 → 퐴 + 푉 + 퐴)
+푑푦
+= 푘+ 푑푁훾∕퐴(푘+)
+푑푘
+휎훾퐴→푉 퐴(푘+)
++ 푘− 푑푁훾∕퐴(푘−)
+푑푘
+휎훾퐴→푉 퐴(푘−),
+(11)
+where 푑푁훾∕퐴∕푑푘 is the photon flux in nucleus 퐴 and 푘 is the
+photon momentum. For fixed rapidity 푦 and transverse momen-
+tum 푝2
+푇 ≈ |푡| of the produced mesons, the photon momentum
+is given by 푘± =
+푀2
+푉 −푡
+2푀푇 푒∓푦 . Here, 푀푇 =
+√
+푀2
+푉 + 푝2
+푇 is the
+transverse mass of the mesons.
+For simplicity, the analytical expression for the flux of pho-
+tons produced by a fast-moving point-like charge has been
+considered (S. R. Klein et al., 2017),
+푑푁훾∕퐴(푘)
+푑푘
+= 2푍2훼푒푚
+휋푘
+[
+푥퐾0(푥)퐾1(푥) − 푥2
+2
+(퐾2
+1(푥) − 퐾2
+0(푥))]
+,
+(12)
+where 푥 = 2푅퐴푘∕훾퐿 and 훾퐿 is the Lorentz factor. 퐾0,1(푥) are
+the modified Bessel functions of the second kind.
+In Fig. 2 results are shown for 휌 production in PbPb, XeXe
+and OO UPCs at the LHC in the rapidity range |푦| ≤ 6. Left
+panel: predictions are presented for the PbPb collisions in ener-
+gies of √푠NN = 5.02 (solid line) and 5.52 TeV (dashed line),
+respectively. It is shown also the measurement performed by
+ALICE Collaboration at mid-rapidity (Acharya et al., 2020).
+Central panel: predictions for XeXe collisions in √푠NN = 5.44
+(solid line) and 5.86 TeV (dashed line) compared to ALICE
+data (Acharya et al., 2021). Right panel: predictions for OO
+collisions with energies of √푠NN = 5.52 (solid line) and 7.00
+TeV (dashed line), respectively. The second energy bin corre-
+sponds to the designed √푠NN for the future High-Luminosity
+LHC (HL-LHC) run (Bruce et al., 2020). In general, the model
+is suitable to predict the magnitude and shape of the rapidity
+distribution in XeXe UPCs. The corresponding suppression at
+central rapidities in PbPb case is consequence of the coherent
+cross section to be underestimated as shown in Fig. 1 Namely,
+the nuclear effects for xenon nucleus are less intense as for lead
+nucleus.
+
+800
+ALICE PbPb UPC 5.02 TeV
+ALICE XeXe UPC 4.44 TeV
+O0 UPC 5.52 TeV
+700
+PbPb UPC 5.02 TeV
+XeXe UPC 5.44 TeV
+DO UPC7.00TeV
+PbPb UPC 5.52 TeV
+180
+XeXeUPC 5.86 TeV
+pAA) [mb]
+0.7
+600
+160
+500
+140
+0.6
+1
+(AA
+300
+120
+do/dy
+0.5
+200
+100
+100
+0.4
+80
+6
+6
+0
+6
+y
+y5
+FIGURE 3 Rapidity distributions for the exclusive 휙 meson photoproduction in ultraperipheral PbPb (left panel), XeXe (central
+panel and OO (right panel) collisions considering the Reggeometric Pomeron model. Prediction are done for the current run
+(solid lines) and future HL-LHC run (dashed lines).
+The contribution of Reggeons to 휌 coherent production turns
+out to be evident in the rapidity distributions at large |푦|. It is
+also 퐴-dependent where the contribution is more important for
+light nuclei than heavy ones. In particular, the energy depen-
+dence of the photon-nucleon cross section, the suppression due
+to nuclear shadowing, and the drop of the flux of high-energy
+photons drive the distribution in the central and forward (back-
+ward) rapidity regions. Bumps or shoulders at large rapidities
+are due to an enhanced contribution of low-energy photopro-
+duction related to the secondary Reggeon exchange in the
+meson-nucleon interaction. The Glauber shadowing at low
+energies is more intense for lead nuclei compared to xenon and
+oxigen ones. This is the reason for the shoulder appearing in
+XeXe and OO collisions and not in PbPb.
+Finally, in Fig. 3 predictions for coherent 휙 photoproduc-
+tion are presented. Using the same notation as previous figure,
+calculations are performed for PbPb, XeXe and OO collisions
+for the energies of the present LHC run (solid lines) and the
+HL-LHC run (dashed lines). Currently, there is no data avail-
+able for 휙 production in AA UPCs at the LHC. It is planned
+a high-granularity detector named FoCal (Bylinkin, Nystrand,
+& Tapia Takaki, 2022) to be installed at the ALICE experi-
+ment, covering large rapidities. It will allow to measure the
+cross sections and expected yields for exclusive production in
+the dielectron decay channel with a coverage for both electrons
+within 3.4 ≤ 휂 ≤ 5.8. The detector FoCal can contribute for
+precise measurements of low-mass vector mesons production
+such as 휌 and 휙 as well as excited 휌 meson states.
+Finally, we discuss the theoretical uncertainties on the cal-
+culations. The predictions are in agreement with those from the
+STARLIGHT Monte Carlo generator for UPCs (S. R. Klein et
+al., 2017) but the cross sections of the coherent meson produc-
+tion are considerably smaller than calculations in Refs. (Frank-
+furt, Guzey, Strikman, & Zhalov, 2016; Guzey, Kryshen, &
+Zhalov, 2020). The main sources of discrepancies are the use
+of the factorized form, Eq. (8), and the classical Glauber for-
+mula, Eq. (6). It is considered the inelastic meson–nucleus
+cross section instead of the total cross section which decreases
+the prediction for the forward cross section by a factor ∼
+2. Namely, the total cross section of the 푉 퐴 interaction
+is obtained from classical mechanics (MC) Glauber model.
+However, the quantum mechanics expression is given by the
+Gribov-Glauber (GG) formalism where the 푉 퐴 cross section
+is given by:
+휎GG
+푡표푡 (푉 퐴) = 2 ∫ 푑2⃗푏
+[
+1 − exp
+(
+−1
+2휎푉 푁푇퐴(⃗푏)
+)]
+.
+(13)
+
+30
+PbPb UPC 5.02 TeV
+XeXe UPC 5.44 TeV
+OO UPC 5.52TeV
+PbPbUPC5.52TeV
+XeXeUPC5.86TeV
+OO UPC 7.00 TeV
+40
+16
+0.08
+do/dy (AA →ΦAA) [mb]
+12
+0.06
+20
+0.04
+10
+0.02
+66
+For example, in the simplification of a sharp sphere nucleus
+with 휌0 = 0.17 fm−3 and radius 푅퐴 one can obtain an estimate
+of the ratio between the GG and CM cross sections,
+휎GG
+푡표푡 (푉 퐴)
+휎CM
+푡표푡 (푉 퐴)
+≈ 2
+(
+1 −
+3
+2휌2
+0휎2
+푉 푁푅2
+퐴
+)
+.
+(14)
+Let us consider the 휌 production. The ratio is ≈ 1.67 for lead
+and 1.55 for xenon by using 휎휌푁 ≈ 25 mb. The classical prob-
+abilistic formula (CM) and the Glauber-Gribov (GG) approach
+give near values of the 휎푡표푡(푉 퐴) only when 휎푡표푡(푉 푝)푇퐴(푏) ≪ 1.
+It is expected that difference for 휙 production be smaller due
+to the lower 휎푡표푡(휙퐴) cross section.
+4
+CONCLUSIONS
+In this contribution predictions for exclusive light vector
+meson photoproduction in UPCs collisions at the LHC are
+presented and compared with the current experimental mea-
+surements. The theoretical approach is based on Regge phe-
+nomenology. In particular, the single-component Reggeomet-
+ric Pomeron model has been considered. Concerning the rapid-
+ity distributions for 휌 production in PbPb UPCs, the model
+underestimates the data whereas does a better job in case of
+XeXe UPCs. Predictions are provided for OO UPCs in a future
+LHC run in light heavy ion model. The results for 휙 follow the
+same trend observed in 휌 production. The theoretical uncer-
+tainties are considerably large concerning the computation of
+nuclear effects, factorization between energy and momentum
+transfer dependence among others.
+The main focus was on the investigation of how models of
+vector meson production in electron-proton scattering affect
+the results in ultra-peripheral nucleus-nucleus collisions. This
+direction of research is especially promising also because of
+the planned experiments at future accelerators. It is promis-
+ing the coverage of the ALICE FoCal detector which allows to
+study the vector meson photoproduction in both low and high
+photon-nucleon centre-of-mass energies.
+ACKNOWLEDGMENTS
+This work was partially supported by the National Academy
+of Science of Ukraine grant 1230/22-1 Fundamental Prop-
+erties of Matter, the Coordination for the Improvement
+of Higher Education Personnel (CAPES/Brazil) grant
+Finance Code 001 and by the National Council for Scien-
+tific and Technological Development (CNPq/Brazil) grant
+306101/2018-1.
+Financial disclosure
+None reported.
+Conflict of interest
+The authors declare no potential conflict of interests.
+REFERENCES
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+

L9E4T4oBgHgl3EQfiw1J/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf,len=467
+page_content='Received ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  Revised ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  Accepted  DOI: xxx/xxxx  PROCEEDINGS IWARA 2022  Light vector meson photoproduction in ultraperipheral heavy ion  collisions at the LHC within the Reggeometric Pomeron approach  László Jenkovszky1 | Érison S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Rocha2 | Magno V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Machado*2  1Bogolyubov ITP, National Academy of  Sciences of Ukraine, Kiev 03143, Ukraine  2HEP Phenomenology Group, Instituto de  Física UFRGS, RS, Brazil  Correspondence  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Machado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Email:  magnus@if.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='ufrgs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='br  Funding Information  National Academy of Science of  Ukraine, 1230/22-1 Fundamental  Properties of Matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Coordination  for the Improvement of Higher Edu-  cation Personnel (CAPES/Brazil),  Finance Code 001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' National Coun-  cil for Scientific and Technolog-  ical Development (CNPq/Brazil),  306101/2018-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  By using the Reggeometric Pomeron model for vector meson production which  successfully describes the high energy lepton-nucleon data, we analyse the light  meson production in ultra-peripheral heavy ion collisions at the Large Hadron Col-  lider (LHC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' The rapidity distributions for 휌 and 휙 photoproduction in lead-lead,  xenon-xenon and oxygen-oxygen collisions are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  KEYWORDS:  ultra-peripheral heavy ion collisions, vector meson photoproduction, Regge phenomenology, Large  Hadron Collider  1  INTRODUCTION  The exclusive light vector meson (푉 ) photoproduction has  been studied in recent years both experimentally and theoreti-  cally (Acharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=', 2020, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Andreev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Klein  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Sirunyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' The process has not asso-  ciated hard perturbative Quantum Chromodynamics (pQCD)  scale in the photoproduction limit, 푄2 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Here, 푄2 is the so  called photon virtuality in the process 훾∗+푁 → 푉 +푁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' There-  fore, light mesons can be used to test the non-perturbative  regime of the strong interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Within the parton satura-  tion formalism the transition between the region described by  pQCD and the non-perturbative regime is interpreted in terms  of the nucleon QCD saturation scale (Morreale & Salazar,  2021), 푄푠(푥) ∼ 푥−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='3, with 푥 being the invariant Bjorken  kinematic variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' In the vector meson electroproduction off  nucleon target, 푥 = (푀2  푉 + 푄2)∕(푊 2  훾푁 + 푄2), where 푊훾푁 is  the centre-of-mass energy of the photon-nucleon system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' In the  QCD color dipole picture (Chen & Mueller, 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Nemchik,  Nikolaev, Predazzi, & Zakharov, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Nikolaev & Zakharov,  1994) 푄푠 characterizes the boundary on the maximum phase-  space gluon density to be reached in the wave-function of the  nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' In this framework, the light meson photoproduction  dynamics at the present accelerator energies can be treated per-  turbatively as 푄푠 reaches values ≲ 1 GeV at very high energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  The perturbative description is even improved in case of scat-  tering off nuclei of atomic number 퐴 as the nuclear saturation  scale, 푄2  푠,퐴 ∝ 퐴1∕3푄2  푠, is enhanced regarding the proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' On  the other hand, it is well known that this approach has been  unable to describe precisely the total photoproduction cross  section and 휌 production at 푄2 = 0 GeV2 (Forshaw, Kerley,  & Shaw, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Gonçalves & Moreira, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' In fact, non-  perturbative corrections are necessary and they are embedded  in the photon wave-function, 휓훾  푇 , for color dipoles containing  large transverse size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  In the soft physics sector, the Regge phenomenology  (Jenkovszky, Schicker, & Szanyi, 2018) is a well founded and  appropriated formalism to describe exclusive diffractive pro-  cesses, including the light meson photoproduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' The vector  meson production amplitude is written in a Regge-factorized  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='05136v1  [hep-ph]  12 Jan 2023  2  structure with the corresponding coupling of particles to the  Pomeron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' The introduction of a perturbative scale depen-  dence suitable for electroproduction can be constructed based  on geometric arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' The Reggeometric Pomeron (RP)  model (Fazio, Fiore, Jenkovszky, & Salii, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Fazio, Fiore,  Lavorini, Jenkovszky, & Salii, 2013) is one example of such  a class of phenomenological models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' The RP model does a  good job in describing both photo and electroproduction at  the DESY-HERA energy regime considering a nucleon target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  The possibility for testing these models in the coherent vec-  tor meson production in ultraperipheral heavy ion collisions  (UPCs) is a reality nowadays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' The basic argument is that the  production cross section in nucleus-nucleus (퐴퐴) collisions  can be factorized in terms of the equivalent flux of photons of  the colliding nucleus and the photon-target production cross  section (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Klein & Steinberg, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  In this contribution, the light meson photoproduction in  nucleus-nucleus UPCs collisions is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' The focus is  on the energies and nuclear species in the heavy ion collisions  at the Large Hadron Collider (LHC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' The theoretical input is  the description based on the Reggeometric Pomeron model for  the elastic differential and integrated total cross section in the  (quasi-real) photon interaction with nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' The parameters  of the model are consistent with the measurements performed  by HERA-H1 (Andreev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=', 2020) and CMS collabora-  tions (Sirunyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=', 2019) as shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' (Jenkovszky,  Rocha, & Machado, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' The nuclear coherent cross section  is then obtained by using Vector Dominance Model (VDM)  and Glauber multiple scattering theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Predictions are per-  formed for 휌 and 휙 production in 퐴퐴 UPCs at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' In  particular, results are compared to the measurements in PbPb  and XeXe UPCs done by ALICE Collaboration (Acharya et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=', 2020, 2021) for the energies of √푠NN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='02 TeV and  √푠NN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='44 TeV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Theoretical estimates for the  cross section in OO collisions are also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' The study is  based on earlier works (Jenkovszky, Libov, & Machado, 2022a,  2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Jenkovszky, Rocha, & Machado, 2022) by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  The work has been organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' 2 we shortly  review the exclusive vector meson production, 훾+푝 → 푉 +푝, in  the context of the Reggeometric Pomeron model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Afterwards,  using VDM model and Glauber formalism for nuclear shad-  owing, the expression for the coherent nuclear cross section is  obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' In section 3 the calculations are compared to avail-  able experimental measurements in PbPb and XeXe UPCs  collisions at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Prediction are done for future light ion  runs like oxygen-oxygen collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Furthermore, discussion  on the theoretical uncertainties is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' In section 4 the  key results are summarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  2  THEORETICAL FRAMEWORK  Exclusive vector meson photoproduction process, 훾 + 푝 →  푉 + 푝, will be described by using a model based on Regge  phenomenology, namely the Reggeometric Pomeron model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' It  is also able to describe electroproduction data as discussed in  what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' In general case, the hardness scale is given by  ̃푄2 = 푄2 + 푀2  푉 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  The elastic differential cross section, 푑휎푒푙∕푑푡, related to the  single-component Reggeometric model in a given scale ̃푄2 is  given by (Fazio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=', 2014, 2013):  푑휎푒푙  푑푡  =  퐴2  0 exp  [  퐵0(̃  푄2) 푡  ]  (  1 +  ̃  푄2  푄2  0  )2푛  (푊 2  훾푝  푊 2  0  )2(훼(푡)−1)  ,  (1)  퐵0(̃  푄2) = 4  (  푎  ̃  푄2 +  푏  2푚2  푁  )  ,  (2)  where the quantity 퐵0(̃  푄2) reflects the geometrical nature of  the model and 훼(푡) denotes the effective Pomeron (퐼푃) trajec-  tory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' The first and second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' (2) correspond to the  effective sizes of the 훾퐼푃 푉 and 푝퐼푃 푝 vertices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' In  the formula above, 푊0 = 1 GeV and 푚푁 is the nucleon mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  It is assumed a linear Pomeron trajectory, 훼(푡) = 훼0 +훼′푡, with  an effective Pomeron intercept 훼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  Accordingly, the integrated cross section is written as,  휎(훾∗ + 푝 → 푉 + 푝) =  퐴2  0  (  1 +  ̃푄2  ̃푄2  0  )2푛  (푊훾푝∕푊0  )4(훼0−1)  퐵  (  푊훾푝, ̃푄2  )  , (3)  퐵  (  푊훾푝, ̃푄2)  = 퐵0(̃  푄2) + 4훼′ ln  (푊훾푝  푊0  )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  (4)  In the photoproduction limit one has ̃푄2 = 푀2  푉 and the param-  eters of the model for 휌 and 휙 production are presented in Table  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' They have been determined (Fazio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=', 2014) by using  DESY-HERA measurements (Aaron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Aid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=',  1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Breitweg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=', 1998, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Derrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=', 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  Now the expressions for the nuclear coherent cross sections  are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Following the STARLIGHT Monte Carlo gen-  erator approach for UPCs processes (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Klein, Nystrand,  Seger, Gorbunov, & Butterworth, 2017), nuclear effects for the  process, 훾 + 퐴 → 푉 + 퐴 are described here by vector domi-  nance model (Bauer, Spital, Yennie, & Pipkin, 1978) and the  classical mechanics Glauber formula for multiple scattering of  the vector meson in the nuclear medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' At 푡 = 0 the differen-  tial cross section is obtained by using the Optical theorem for  scattering in a nucleus and VDM as follows,  d휎 (훾 + 퐴 → 푉 + 퐴)  d푡  ||||푡=0  = 훼푒푚  4푓 2  푉  휎2  푡표푡 (푉 퐴) ,  (5)  휎푡표푡 (푉 퐴) = ∫ d2b [1 − 푒−휎푡표푡(푉 푝)푇퐴(b)] ,(6)  3  TABLE 1 Values of the parameters for the Reggeometric Pomeron model (Fazio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  Meson  퐴0  [ √  nb  GeV  ]  ̃  푄2  0  [GeV2]  푛  훼0  훼′ [GeV−2]  푎  푏  휌  344 ± 376  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='14  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='07  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='53  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='33  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='9 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='3  휙  58 ± 112  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='89 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='30± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='28  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='14± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='19  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='78  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='0 ± 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='34± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='09  where 푇퐴(푏) is the nuclear thickness function and 푓푉 is  the vector-meson coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' The values 푓 2  휌 ∕4휋 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='02 and  푓 2  휙∕4휋 = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='7 are considered in calculations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' For  light mesons, 휎푡표푡(푉 푝) is large and the cross section 휎푡표푡(푉 퐴)  is approximately the geometric cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' It is also almost  energy independent (Jenkovszky, Rocha, & Machado, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  The input for the Glauber model calculation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' (6) is the  effective vector meson–nucleon cross for the process 푉 + 푝 →  푉 + 푝, which is given by:  휎푡표푡 (푉 푝) =  √  4푓 2  푉  훼푒푚  d휎 (훾 + 푝 → 푉 + 푝)  d푡  ||||푡=0  ,  (7)  where the differential cross section coming from the Reggeo-  metric Pomeron model, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' (1), will be introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' (7) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  The corresponding integrated cross section is given by:  휎(훾 + 퐴 → 푉 + 퐴) = 푑휎(훾 + 퐴 → 푉 + 퐴)  푑푡  ||||푡=0  ×  ∞  ∫  푡푚푖푛  d|푡| ||퐹퐴 (푡)||  2 ,  (8)  where the quantity 퐹퐴 is the nuclear form factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' It is taken  into account an analytic form factor given by a hard sphere  of radius, 푅퐴 = 푟0퐴1∕3 fm (푟0 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='2 fm), convoluted with a  Yukawa potential with range 푎 (Davies & Nix, 1976),  퐹퐴(|푞|) = 4휋휌0  퐴|푞3|  (  1  1 + 푎2푞2  )  × [sin (|푞|푅퐴) − |푞|푅퐴 cos (|푞|푅퐴)] ,  (9)  where 푞 is the momentum transfer, 휌0 = 3퐴∕(4휋푅3  퐴) fm−3 and  푎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='7 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  In the calculations the reggeon contribution is added to  the photoproduction off nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' The corresponding cross  section is parameterized as,  푑휎퐼푅(훾푝 → 푉 푃)  푑푡  |||||푡=0  = 푏푉 푌 푊 −휂  훾푝 ,  (10)  where the constants 푏푉  = 11 GeV−2, 푌  = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='0 휇b and  휂 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='23 have been considered for the 휌 production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' For the  휙, meson exchange is strongly suppressed, and the reaction  occurs only through 퐼푃-exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  The 퐴-dependence of the cross section for the coherent pro-  duction of 휌 meson, 휎(훾 + 퐴 → 휌 + 퐴), is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  FIGURE 1 The 퐴-dependence of the cross section for the  coherent production of 휌 meson from Reggeometric Pomeron  model at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Data from ALICE Collaboration (Acharya  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Comparison is done with the extracted values of the coher-  ent cross section performed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' (Acharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=', 2021) by  ALICE Collaboration using the measured data on UPCs at the  LHC (PbPb collisions at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='02 TeV and XeXe at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='44 TeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  The description is quite reasonable for the nuclear dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  At central rapidity, 푦 = 0, the photon-nucleon centre-of-mass  energy squared is 푊 2  훾푁 = 푀푉  √푠NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' For xenon the cross  section corresponds to 푊훾푁 = 65 GeV and 휎(훾Xe → 휌Xe) ≃  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='21 mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' The predicted values from the Reggeomet-  ric Pomeron model is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='07 mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' For lead, the data is 휎(훾Pb →  휌Pb) ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='16 mb for energy 푊훾푁 = 62 GeV and the  prediction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='54 mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Theoretical calculation underestimates the  extracted 훾Pb cross section, which suggests a strong nuclear  shadowing correction for very large nucleus in the formalism  considered for the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  3,0  ALICE-Xe  2,5  ALICE-Pb  ReggeometricPomeron model  [mb]  +A)  P1,5  个  V+i)  1,0  a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='5  0,0  1  0  20  40  60  80  100  120  140  160  180  200  220  240  A4  FIGURE 2 Rapidity distributions for the exclusive 휌 meson photoproduction in ultraperipheral PbPb (left panel), XeXe (central  panel and OO (right panel) collisions considering the Reggeometric Pomeron model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Prediction are done for the current run  (solid lines) and future HL-LHC run (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Comparison is done to ALICE Collaboration data (Acharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=', 2020,  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  3  RESULTS AND DISCUSSIONS  The rapidity distribution for meson production in nucleus-  nucleus UPCs takes a factorized form in the Equivalent Photon  Approximation (EPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' The expression is given by:  푑휎(퐴 + 퐴 → 퐴 + 푉 + 퐴)  푑푦  = 푘+ 푑푁훾∕퐴(푘+)  푑푘  휎훾퐴→푉 퐴(푘+)  + 푘− 푑푁훾∕퐴(푘−)  푑푘  휎훾퐴→푉 퐴(푘−),  (11)  where 푑푁훾∕퐴∕푑푘 is the photon flux in nucleus 퐴 and 푘 is the  photon momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' For fixed rapidity 푦 and transverse momen-  tum 푝2  푇 ≈ |푡| of the produced mesons, the photon momentum  is given by 푘± =  푀2  푉 −푡  2푀푇 푒∓푦 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Here, 푀푇 =  √  푀2  푉 + 푝2  푇 is the  transverse mass of the mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  For simplicity, the analytical expression for the flux of pho-  tons produced by a fast-moving point-like charge has been  considered (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Klein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=', 2017),  푑푁훾∕퐴(푘)  푑푘  = 2푍2훼푒푚  휋푘  [  푥퐾0(푥)퐾1(푥) − 푥2  2  (퐾2  1(푥) − 퐾2  0(푥))]  ,  (12)  where 푥 = 2푅퐴푘∕훾퐿 and 훾퐿 is the Lorentz factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' 퐾0,1(푥) are  the modified Bessel functions of the second kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' 2 results are shown for 휌 production in PbPb, XeXe  and OO UPCs at the LHC in the rapidity range |푦| ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Left  panel: predictions are presented for the PbPb collisions in ener-  gies of √푠NN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='02 (solid line) and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='52 TeV (dashed line),  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' It is shown also the measurement performed by  ALICE Collaboration at mid-rapidity (Acharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  Central panel: predictions for XeXe collisions in √푠NN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='44  (solid line) and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='86 TeV (dashed line) compared to ALICE  data (Acharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Right panel: predictions for OO  collisions with energies of √푠NN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='52 (solid line) and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='00  TeV (dashed line), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' The second energy bin corre-  sponds to the designed √푠NN for the future High-Luminosity  LHC (HL-LHC) run (Bruce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' In general, the model  is suitable to predict the magnitude and shape of the rapidity  distribution in XeXe UPCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' The corresponding suppression at  central rapidities in PbPb case is consequence of the coherent  cross section to be underestimated as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' 1 Namely,  the nuclear effects for xenon nucleus are less intense as for lead  nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  800  ALICE PbPb UPC 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='02 TeV  ALICE XeXe UPC 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='44 TeV  O0 UPC 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='52 TeV  700  PbPb UPC 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='02 TeV  XeXe UPC 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='44 TeV  DO UPC7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='00TeV  PbPb UPC 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='52 TeV  180  XeXeUPC 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='86 TeV  pAA) [mb]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='7  600  160  500  140  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='6  1  (AA  300  120  do/dy  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='5  200  100  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='4  80  6  6  0  6  y  y5  FIGURE 3 Rapidity distributions for the exclusive 휙 meson photoproduction in ultraperipheral PbPb (left panel), XeXe (central  panel and OO (right panel) collisions considering the Reggeometric Pomeron model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Prediction are done for the current run  (solid lines) and future HL-LHC run (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  The contribution of Reggeons to 휌 coherent production turns  out to be evident in the rapidity distributions at large |푦|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' It is  also 퐴-dependent where the contribution is more important for  light nuclei than heavy ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' In particular, the energy depen-  dence of the photon-nucleon cross section, the suppression due  to nuclear shadowing, and the drop of the flux of high-energy  photons drive the distribution in the central and forward (back-  ward) rapidity regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Bumps or shoulders at large rapidities  are due to an enhanced contribution of low-energy photopro-  duction related to the secondary Reggeon exchange in the  meson-nucleon interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' The Glauber shadowing at low  energies is more intense for lead nuclei compared to xenon and  oxigen ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' This is the reason for the shoulder appearing in  XeXe and OO collisions and not in PbPb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  Finally, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' 3 predictions for coherent 휙 photoproduc-  tion are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Using the same notation as previous figure,  calculations are performed for PbPb, XeXe and OO collisions  for the energies of the present LHC run (solid lines) and the  HL-LHC run (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Currently, there is no data avail-  able for 휙 production in AA UPCs at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' It is planned  a high-granularity detector named FoCal (Bylinkin, Nystrand,  & Tapia Takaki, 2022) to be installed at the ALICE experi-  ment, covering large rapidities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' It will allow to measure the  cross sections and expected yields for exclusive production in  the dielectron decay channel with a coverage for both electrons  within 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='4 ≤ 휂 ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' The detector FoCal can contribute for  precise measurements of low-mass vector mesons production  such as 휌 and 휙 as well as excited 휌 meson states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  Finally, we discuss the theoretical uncertainties on the cal-  culations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' The predictions are in agreement with those from the  STARLIGHT Monte Carlo generator for UPCs (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Klein et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=', 2017) but the cross sections of the coherent meson produc-  tion are considerably smaller than calculations in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' (Frank-  furt, Guzey, Strikman, & Zhalov, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Guzey, Kryshen, &  Zhalov, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' The main sources of discrepancies are the use  of the factorized form, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' (8), and the classical Glauber for-  mula, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' It is considered the inelastic meson–nucleus  cross section instead of the total cross section which decreases  the prediction for the forward cross section by a factor ∼  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Namely, the total cross section of the 푉 퐴 interaction  is obtained from classical mechanics (MC) Glauber model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  However, the quantum mechanics expression is given by the  Gribov-Glauber (GG) formalism where the 푉 퐴 cross section  is given by:  휎GG  푡표푡 (푉 퐴) = 2 ∫ 푑2���푏  [  1 − exp  (  −1  2휎푉 푁푇퐴(⃗푏)  )]  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  (13)  30  PbPb UPC 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='02 TeV  XeXe UPC 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='44 TeV  OO UPC 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='52TeV  PbPbUPC5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='52TeV  XeXeUPC5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='86TeV  OO UPC 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='00 TeV  40  16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='08  do/dy (AA →ΦAA) [mb]  12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='06  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='04  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='02  66  For example, in the simplification of a sharp sphere nucleus  with 휌0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='17 fm−3 and radius 푅퐴 one can obtain an estimate  of the ratio between the GG and CM cross sections,  휎GG  푡표푡 (푉 퐴)  휎CM  푡표푡 (푉 퐴)  ≈ 2  (  1 −  3  2휌2  0휎2  푉 푁푅2  퐴  )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  (14)  Let us consider the 휌 production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' The ratio is ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='67 for lead  and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='55 for xenon by using 휎휌푁 ≈ 25 mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' The classical prob-  abilistic formula (CM) and the Glauber-Gribov (GG) approach  give near values of the 휎푡표푡(푉 퐴) only when 휎푡표푡(푉 푝)푇퐴(푏) ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  It is expected that difference for 휙 production be smaller due  to the lower 휎푡표푡(휙퐴) cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  4  CONCLUSIONS  In this contribution predictions for exclusive light vector  meson photoproduction in UPCs collisions at the LHC are  presented and compared with the current experimental mea-  surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' The theoretical approach is based on Regge phe-  nomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' In particular, the single-component Reggeomet-  ric Pomeron model has been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Concerning the rapid-  ity distributions for 휌 production in PbPb UPCs, the model  underestimates the data whereas does a better job in case of  XeXe UPCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' Predictions are provided for OO UPCs in a future  LHC run in light heavy ion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' The results for 휙 follow the  same trend observed in 휌 production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' The theoretical uncer-  tainties are considerably large concerning the computation of  nuclear effects, factorization between energy and momentum  transfer dependence among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  The main focus was on the investigation of how models of  vector meson production in electron-proton scattering affect  the results in ultra-peripheral nucleus-nucleus collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' This  direction of research is especially promising also because of  the planned experiments at future accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content=' It is promis-  ing the coverage of the ALICE FoCal detector which allows to  study the vector meson photoproduction in both low and high  photon-nucleon centre-of-mass energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This work was partially supported by the National Academy  of Science of Ukraine grant 1230/22-1 Fundamental Prop-  erties of Matter, the Coordination for the Improvement  of Higher Education Personnel (CAPES/Brazil) grant  Finance Code 001 and by the National Council for Scien-  tific and Technological Development (CNPq/Brazil) grant  306101/2018-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  Financial disclosure  None reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
+page_content='  Conflict of interest  The authors declare no potential conflict of interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
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+page_content='  Bylinkin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
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+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf'}
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+arXiv:2301.00113v1  [astro-ph.HE]  31 Dec 2022
+Black hole images: A Review
+Songbai Chen 1,2 ∗, Jiliang Jing 1,2, Wei-Liang Qian 2,3,4, Bin Wang 2,5
+1 Department of Physics, Synergetic Innovation Center for Quantum Effects and Applications,
+Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China
+2 Center for Gravitation and Cosmology, College of Physical Science and Technology,
+Yangzhou University, Yangzhou 225009, People’s Republic of China
+3 Escola de Engenharia de Lorena, Universidade de S˜ao Paulo, 12602-810, Lorena, SP, Brazil
+4 Faculdade de Engenharia de Guaratinguet´a,
+Universidade Estadual Paulista, 12516-410, Guaratinguet´a, SP, Brazil
+5 School of Aeronautics and Astronautics, Shanghai Jiao Tong University,
+Shanghai 200240, People’s Republic of China
+In recent years, unprecedented progress has been achieved regarding black holes’ observation
+through the electromagnetic channel. The images of the supermassive black holes M87∗ and Sgr
+A∗ released by the Event Horizon Telescope (EHT) Collaboration provided direct visual evidence
+for their existence, which has stimulated further studies on various aspects of the compact celestial
+objects. Moreover, the information stored in these images provides a new way to understand the
+pertinent physical processes that occurred near the black holes, to test alternative theories of gravity,
+and to furnish insight into fundamental physics. In this review, we briefly summarize the recent
+developments on the topic. In particular, we elaborate on the features and formation mechanism of
+black hole shadows, the properties of black hole images illuminated by the surrounding thin accretion
+disk, and the corresponding polarization patterns. The potential applications of the relevant studies
+are also addressed.
+Key words: black hole shadow, accretion disk, polarization image
+PACS numbers: 04.70.Cs, 98.62.Mw, 97.60.Lf
+I.
+INTRODUCTION
+The releasing of the first image of the supermas-
+sive black hole M87∗ by the EHT Collaboration in
+2019 [1–6] is a milestone event in physics. It provided
+direct visual evidence of black hole in our universe,
+which means that black hole is no longer just a theo-
+retical model. Combining with the recently published
+black hole image of Sgr A∗ [7], it is widely believed
+that the observational astronomy of black holes has
+entered a new era of rapid progress.
+One of the most important ingredients in the im-
+ages is the black hole shadow [8].
+It is a two di-
+mensional dark region in the observer’s sky, which is
+caused by light rays falling into an event horizon of
+a black hole [9–12]. The captured light rays are very
+close to the black hole so that the shape and size of
+the shadow carry the fingerprint of the celestial ob-
+ject. Therefore, the study of shadows is beneficial to
+identify black holes, to examine theories of gravity in-
+cluding general relativity, and to further understand
+some fundamental problems in physics.
+From the light propagation in spacetime, black
+hole shadow depends on the light source, the back-
+ground spacetime and even the properties of electro-
+dynamics obeyed by photon itself. The light sources
+in many theoretical investigations are assumed to ho-
+mogeneously distribute in the total celestial sphere.
+However, in the real astronomical environment, ac-
+∗Corresponding author: [email protected]
+cretion disk is a kind of actual and feasible light
+sources around black holes due to its electromagnetic
+emission. Undoubtedly, the matter configuration and
+the accretion process in the disks are indelibly im-
+printed on black hole images. Conversely, analyzing
+the luminosity distribution and electromagnetic sig-
+nals stored in the images can extract the informa-
+tion about the matter fields and the physical pro-
+cesses near the black holes.
+The twisting patterns
+in the first polarized image of the M87* black hole
+[13, 14] revealed the presence of a poloidal magnetic
+field about 1 ∼ 30G near the black hole. Thus, black
+hole images with polarized information provide an-
+other new way to probe the matter distribution, the
+electromagnetic interactions and the accretion pro-
+cesses in the strong gravity region of black holes.
+The literature related to black hole images is
+rapidly increasing. Therefore, it is necessary to sum-
+marize the existing works at the present time and
+make prospects for the future. This review focuses
+on the black hole images, and their rapid develop-
+ment and potential applications. We first introduce
+the basic concepts of black hole shadows and sum-
+marize the main features of the shadows and their
+formation mechanism, and review how the black hole
+shadows are determined by the fundamental photon
+orbits [15] and the corresponding invariant manifolds
+[16]. Next, we introduce the images of the black holes
+surrounded by a thin disk and their polarization pat-
+terns arising from synchrotron radiation.
+We also
+discuss the aspects of the potential applications of
+black hole images.
+
+2
+observer
+photon sphere
+rsh
+FIG. 1: The event horizon (the black disk), the photon
+sphere rps = 3M (the red circle) and the shadow with a
+radius rsh = 3
+√
+3M for a Schwarzschild black hole [22].
+II.
+BLACK HOLE SHADOWS AND THEIR
+FORMATION
+It is well known that the shadow of a common ob-
+ject is determined by the light rays passing through
+the edge of the object. However, for the objects with
+strong gravity, such as the black hole, the situation is
+different. According to general relativity, light rays
+travelling in a black hole spacetime can be deflected
+due to the gravitational field of the black hole. This
+phenomenon is known as gravitational lensing, which
+is analogous to optical lensing [17–19]. The deflection
+angle of the light ray increases with the decreasing of
+its impact parameter. Therefore, it is easy to infer
+that the light rays passing very close to the black
+hole will be captured. Black hole shadow is a dark
+silhouette observed in the sky originating from these
+captured light rays. Although the black hole shadow
+is caused by the photons fallen into the event horizon,
+its size is not equal to that of this null hypersurface.
+Actually, for a Schwarzschild black hole, its shadow
+is about 2.5 times as large as the event horizon in
+angular size [9, 10]. This can be explained by two
+reasons. Firstly, it is not only the photons near the
+event horizon can be captured by a black hole. In
+fact, there exists a photon sphere outside the event
+horizon, which is an envelope surface of unstable pho-
+ton circular orbits in the spacetime [20–22]. The light
+rays entered the photon sphere will be captured by
+the black hole as shown in Fig.1. Thus, the bound-
+ary of the shadow is determined by the photon sphere
+rather than the event horizon. Secondly, due to the
+strong bending of light rays induced by black hole’s
+gravity, both the size and shape of the observed dark
+shadow are different from those naively based on Eu-
+clidean geometry without gravity.
+A.
+Features of black hole shadows
+The shape and size of black hole shadows depend
+on the black hole parameters and the observer incli-
+nation. For a Schwarzschild black hole, the shadow
+is a perfect disk for the observer with arbitrary incli-
+nation [10, 12]. For a rotating Kerr black hole, the
+shadow also presents a circular silhouette for the ob-
+server located on its rotation axis. However, for the
+FIG. 2:
+The eyebrowlike shadows near the primary
+shadow for a Kerr black hole with scalar hair [32].
+observer in the equatorial plane, the shadow gradu-
+ally becomes a “D”-shaped silhouette with increasing
+black hole spin [10, 12]. For a Konoplya-Zhidenko ro-
+tating non-Kerr black hole with an extra deformation
+parameter described the deviations from the Kerr
+metric, the “D”-shaped shadow could disappear and
+the special cuspy shadow could emerge in a certain
+range of parameter values for the equatorial observer
+[23].
+This is also true for black holes with Proca
+hair [15, 24]. Thus, the dependence of shadows on
+black hole parameters could provide a potential tool
+to identify black holes in nature. It also triggers the
+further study of black hole shadows in various theo-
+ries of gravity [25–30].
+Black hole shadow also depends on the (non-
+)integrability of motion equation of photon travelling
+in the background spacetime. The completely inte-
+grable systems are such kind of dynamical systems
+where the number of the first integrals is equal to its
+degrees of freedom. Generally, in static spacetimes
+of black holes with spherical symmetry, such as in
+the Schwarzschild black hole spacetime, the dynam-
+ical system of photons is completely integrable since
+it possesses three independent first integrals, i.e., the
+energy E, the z-component of the angular momen-
+tum Lz and the Carter constant Q [31].
+This en-
+sures the motion of photons is regular so that black
+hole shadow has the same shape as the photon sphere
+surface. In these completely integrable systems, the
+black hole shadow can be calculated by analytical
+methods [10, 12].
+However, as the dynamical sys-
+tem of photons is not completely integrable, the null
+geodesic equations are not be variable separable be-
+cause there is no existence of a Carter-like constant
+apart from the usual two integrals of motion E and
+Lz. This implies that the motion of photons could
+be chaotic and sharply affect the shadow so that its
+shape is different from that of the photon sphere sur-
+face [32–37].
+In particular, due to chaotic lensing,
+there are eyebrowlike shadows with the self-similar
+fractal structure near the primary shadow, as shown
+in Fig.2. In addition, the black hole shadows in the
+nonintegrable cases can be only obtained by numeri-
+cal simulations with the so-called “ray-tracing” codes
+[32, 35, 38, 39].
+
+3
+B.
+Formation mechanism of black hole shadows
+The photon sphere plays an important role in the
+formation of black hole shadows. Actually, the pho-
+ton sphere is composed of unstable photon circular
+orbits around black holes. The unstable photon cir-
+cular orbits in the equatorial plane are determined by
+the effective potential and its derivatives [40, 41], i.e.,
+Veff(r) = 0, Veff(r),r = 0 and Veff(r),rr < 0. These
+unstable orbits can also be obtained by a geometric
+way with Gauss curvature and geodesic curvature in
+the optical geometry [42]. For a static four dimen-
+sional spacetime, its optical geometry restricted in
+the equatorial plane is defined by dt2 ≡ gOP
+ij dxidxj,
+i = r, φ. The geodesic curvature and Gauss curvature
+can be expressed as [42]
+κgeo =
+1
+2
+�
+gOP
+rr
+∂ ln gOP
+φφ
+∂r
+,
+(1)
+κGau = −
+1
+�
+gOP
+� ∂
+∂φ
+�
+1
+�
+gOP
+φφ
+∂
+�
+gOP
+rr
+∂φ
+�
++ ∂
+∂r
+�
+1
+�
+gOP
+rr
+∂
+�
+gOP
+φφ
+∂r
+��
+.
+(2)
+The geodesic curvature κgeo = 0 gives the radius of
+the circular photon orbit and the positive (negative)
+of Gauss curvature κGau determines that the circular
+photon orbit is stable (unstable).
+Fundamental photon orbits
+The unstable photon
+circular orbits in the equatorial plane are often called
+light rings. Light rings also affect dynamical proper-
+ties of ultracompact objects [15]. For the ultracom-
+pact objects without horizon [43], light rings often
+come in pairs, one stable and the other unstable. The
+existence of a stable light ring always implies a space-
+time instability [44]. In the Schwarzschild spacetime,
+light rings are the only bound photon orbits. In the
+rotating Kerr spacetime, there are two light rings
+located in the equatorial plane, one for co-rotating
+photon and one for counter-rotating photon with re-
+spect to the black hole. Moreover, there also exist the
+non-planar bound photon orbits with constant r and
+motion in θ, known as spherical orbits ( see also Fig.
+2 in [15]). These spherical orbits are unstable and
+completely determine the Kerr black hole shadow.
+Fundamental photon orbits are the generalization
+of light rings and spherical orbits in usual stationary
+and axisymmetric spacetimes [15]. The definition of
+fundamental photon orbits is given in [15]. According
+to the features of orbits, the fundamental photon or-
+bits can be categorized as Xnr±
+ns
+, where X = {O, C},
+and nr, ns ∈ N0. The orbit O is open and it can reach
+the boundary of the effective potential. The orbit C
+is closed and it can not reach the boundary.
+The
+sign +(−) denotes the even (odd) parity of the orbit
+under the Z2 reflection symmetry around the equa-
+torial plane. nr is the number of distinct r values at
+FIG. 3: Some fundamental photon orbits in the (r, θ)-
+plane and their classification. The grey areas represent
+forbidden regions for the effective potential [15].
+where the orbit crosses the equatorial plane. For the
+light rings lied in the equatorial plane, their nr = 0
+because such special kind of orbits never cross the
+equatorial plane. ns is the number of self-intersection
+points of the orbit. In Fig.3, some fundamental pho-
+ton orbits and their classification are illustrated in
+the (r, θ)-plane.
+By making use of the fundamental photon orbits,
+P. V. P. Cunha et al. [15] explained the formation
+of the cuspy silhouette of a Kerr black hole with
+Proca hair shown in Fig.4(a). To clearly demonstrate
+the formation mechanism of the black hole shadow,
+ten fundamental photon orbits are selected out and
+marked by “A1-A4, B1-B3, C1-C3”. Then, the dis-
+tribution of ∆θ ≡ |θmax − π
+2 | and rperi with the im-
+pact parameter η are presented for each fundamental
+photon orbit, where θmax is the maximal/minimal
+angular coordinate at the spherical orbit and rperi
+is the perimetral radius as a spherical orbit crosses
+the equatorial plane.
+The right panel of Fig.4(b)
+shows the spatial trajectories of the ten fundamental
+photon orbits in Cartesian coordinates, which move
+around the black hole.
+The orbits A1 and C3 are
+the unstable prograde and retrograde light rings re-
+spectively shown as two black circles on the equa-
+torial plane. Other fundamental photon orbits are
+non-planar bound photon orbits crossing the equa-
+torial plane. The continuum of fundamental photon
+orbits can be split into one stable branch (the red dot-
+ted line) and two unstable branches (the green and
+blue lines). The swallow-tail shape pattern related to
+the fundamental photon orbits in the η − ∆θ plane
+yields a jump occurred at A4 and C1 in the η − rperi
+plane. The discontinuity originating from this jump,
+i.e., rperi(C1) > rperi(A4), induces the emergence of
+the cuspy shadow. The unstable fundamental pho-
+ton orbits (C1-B3) non-related to the shadow could
+be associated to a set of lensing patterns attached to
+the shadow edge, called “eyelashes” in Fig.4(a). In
+the black hole spacetimes where there exists a second
+pair of light rings, the fundamental photon orbits can
+be classified into two fully disconnected branches: the
+shadow related branch and the non-shadow related
+one. If the shadow related branch is connected, the
+
+.0
+Ci3+
+04
+(a)
+(b)
+FIG. 4:
+The cuspy shadow (a) and the fundamental pho-
+ton orbits (b) for the Kerr black hole with Proca hair[15].
+shadow edge will be smooth with no cusp. But the
+eyelashes, caused by the non-shadow related unstable
+branch, appear to be disconnected from the shadow,
+forming a pixelated banana-shaped strip in the lens-
+ing image as shown in Fig.2. This feature has been
+dubbed “ghost shadow” in [24]. This mechanism is
+also applied to explain the formation of the cuspy
+shadow in the Konoplya-Zhidenko rotating non-Kerr
+black hole spacetime [23]. It is further confirmed that
+the unstable fundamental photon orbits play an im-
+portant role in determining the boundary of shadow
+and the patterns of the shadow shape. In terms of
+a toy model [45], the feature of cuspy shadow can
+be derived by employing the Maxwell construction
+for phase transition in a two-component system. In
+addition, the shadows for the black holes with gen-
+eral parameterized metrics have been studied by us-
+ing the fundamental photon orbits [46–51].
+These
+studies could also be beneficial to test the Kerr hy-
+pothesis through black hole shadows.
+Invariant phase space structures The invariant
+phase space structures are very important for dynam-
+ical systems because they remain invariant under the
+dynamics. There are several types of invariant struc-
+tures including fixed points, periodic orbits and in-
+variant manifolds. The simplest is the fixed points.
+These phase space structures are applied extensively
+to design space trajectory for various of spacecrafts
+[52–55]. Since black hole shadow depends on the dy-
+namics of photons in the background spacetime, the
+invariant phase space structures should also play an
+important role in the formation mechanism of the
+shadow. For a dynamical system of photons travel-
+ling in a curved spacetime, its fixed points can be
+determined by the conditions
+˙xµ = ∂H
+∂pµ
+= 0,
+˙pµ = − ∂H
+∂xµ = 0,
+(3)
+where qµ = (t, r, θ, ϕ) and pν = (pt, pr, pθ, pϕ), H is
+the Hamiltonian of the system. Actually, the light
+rings on the equatorial plane are the fixed points for
+the photon motion [15, 16].
+In the vicinity of the
+fixed points, one can linearize the equations (3) and
+obtain a matrix equation
+˙X = JX,
+(4)
+where X = (qµ, pν) and J is the Jacobian matrix.
+The eigenvalues µj of the Jacobian matrix J deter-
+mine the local dynamical properties of the system
+near the fixed points (see also in Fig.??). The stable
+and unstable invariant manifolds correspond to the
+cases of Re(µj) < 0 and Re(µj) > 0, respectively;
+while the center manifold corresponds to the case
+with Re(µj) = 0 where the eigenvalues µj are pure
+imaginary numbers. Due to the special properties of
+the invariant manifolds, there is no trajectory cross-
+ing the invariant manifolds. Points in the unstable
+(stable) invariant manifold move to the fixed points
+exponentially in backward (forward) time. In terms
+of Lyapunov’s central limit theorem, the eigenvalue
+with Re(µj) = 0 leads to the so-called Lyapunov or-
+bits, which is a one-parameter family γǫ of periodic
+orbits [16, 52–55]. These orbits γǫ in the center man-
+ifold collapse into a fixed point as ǫ → 0. Similarly,
+the periodic orbits also have their own stable and un-
+stable manifolds. These invariant phase space struc-
+tures are also shown in Fig. 1 in [16]. Obviously, the
+photon spheres and other periodic orbits can be gen-
+eralized to the Lyapunov orbits related to the fixed
+points [16].
+These concepts in dynamical systems provide a
+powerful theoretical foundation for understanding
+the formation of shadows cast by black holes. The
+unstable invariant manifold builds a bridge between
+the photon sphere and the observer because such
+manifold can approach the fixed points exponentially
+in backward time [16, 35]. Only the Lyapunov family
+of the spherical orbits near the unstable fixed points
+are responsible for generating the black hole shadow.
+For a Kerr black hole with scalar hair, there are three
+unstable light rings L1, L2 and L3.
+However, the
+Lyapunov orbits associated with L1 is non-spherical
+and only the orbits emanating from L2 and L3 are
+
+J (W)
+a-
+2-
+4
+-3
+S-
+-
+0
+S
+C3
+TA
+CJ=V
+2.0
+CS
+BJ
+B3'CJ-C3
+A=O
+SA
+BS
+eldsta
+B3
+CA
+0
+TA
+VS
+AA
+EA
+2.0
+BJ
+BS
+bel!
+B3
+a.t
+5
+A-TA
+Cs
+C3......5
+FIG. 5: Intersections of the unstable manifolds of L1, L2,
+and L3 as well as their Lyapunov orbits with the image
+plane.
+Lyapunov orbits related to L1, L2, and L3 are
+marked by red, green, and blue dots, respectively [16].
+spherical [16]. The numerical simulated shadow in
+Fig.5 shows that the complicated and disconnected
+boundaries of the shadow are completely determined
+by the Lyapunov spherical orbits. Thus, the invari-
+ant manifolds of certain Lyapunov orbits are directly
+related to black hole shadows even in the case of com-
+plicated non-convex, disconnected shadows.
+More-
+over, the curved streamlines in the unstable invariant
+manifolds could lead to that the shape of the black
+hole shadow detected by the observer at spatial in-
+finity differs from that of the photon sphere surface
+[16, 35, 37].
+III.
+IMAGE OF A BLACK HOLE WITH A
+THIN ACCRETION DISK
+In the real astrophysical conditions, a black hole
+is surrounded by a hot accretion disk within which
+it emits a characteristic spectrum of electromagnetic
+radiation. The electromagnetic radiation emitted by
+the disk illuminates the background around the black
+hole and makes the black hole shadow visible. Thus,
+the accretion disk is the actual light source in the
+formation of black hole shadows. Clearly, the black
+hole image cast by the light source with the disk-like
+structure differs from that by the former homoge-
+neous light source in the previous analyses. Simulta-
+neously, due to the strong gravitational lensing near
+the central black hole, the shape of the accretion disk
+is heavily distorted.
+Luminet
+first
+simulated
+a
+photograph
+of
+a
+Schwarzschild black hole with a rotating thin accre-
+tion disk [56].
+Here, the proper luminosity of the
+disk is calculated according to the model described
+by Page and Thorne [57] in its relativistic version,
+where the intensity of radiation emitted at arbitrary
+given point of the disk only depends on the radial dis-
+tance to the black hole. As shown in Fig.11 in [56],
+the flying-saucer-shaped bright region is the primary
+image of disk, which is formed by the light emitted
+directly from the upper side of the disk [56].
+Due
+to the considerable distortion caused by the strong
+gravitational lensing near the central black hole, the
+primary image related to the back part of the disk
+is completely visible rather than hidden by the black
+hole.
+Moreover, one also sees a highly deformed image
+associated with the bottom of the gaseous disk. This
+is because the light rays emitted from the bottom side
+can climb back to the top and reach to the observer at
+the spatial distance [56]. Actually, the gravitational
+lensing gives rise to an infinity of images of the disk,
+which are caused by the light rays traveling around
+the black hole any number of times before reaching a
+distant astronomer [56, 58]. The number of times of
+light ray crossing the disk determines the order of the
+image. The higher order images are closer to the cen-
+tral black spot and become thinner and fainter. The
+inner infinite order image is related to the photon
+sphere, which represents the actual shadow boundary
+of the black hole. Generally, it is difficult to distin-
+guish the higher order images optically because they
+are standing quite closely to each other. The central
+black area is the black hole shadow formed by the
+gravitational lensing and capture of light rays.
+The existence of a dark gap between the primary
+image and the higher order images is not surprising
+because the accretion disk is forbidden to touch the
+surface of the black hole and then there is not any ra-
+diation from the region between the black hole’s event
+horizon and the inner edge of the disk. Moreover, be-
+low the inner stable circular orbit, the disk is unstable
+so that the gas particles plunge directly towards the
+black hole without having enough time to emit elec-
+tromagnetic radiation [56]. Unlike the shadow itself,
+the darkness in these patches is of a fundamentally
+different nature, which may be filled with the emis-
+sion from lensed images of distant sources in the en-
+tire universe although it will also be extremely faint
+[59, 60].
+For a disk around the black hole, the region closer
+to the horizon is generally brighter because the gas
+is hotter there.
+However, the apparent luminosity
+of the disk’s image for the distant observer is very
+different from the intrinsic luminosity in the disk.
+The main reason is that the electromagnetic radi-
+ation detected at a great distance undergoes shifts
+in frequency and intensity with respect to the orig-
+inal radiation emitted directly by the disk [56, 61].
+There are two kinds of shift effects.
+One of them
+is the so-called gravitational redshift caused by the
+gravity of the central black hole, which lowers the fre-
+quency and decreases the intensity of the electromag-
+netic radiation. The other is the well-known Doppler
+effect originating from the displacement of the source
+with respect to the observer. Doppler effect gives rise
+to amplification for the approaching source and at-
+tenuation for the retreating source. Therefore, for a
+disk rotating counterclockwise around the black hole,
+the apparent luminosity of the disk in the left side is
+
+000
+a.0-=n
+U=-s'e6
+brighter than that of in the right side [56]. The strong
+gravity of the black hole can give a speed of gas rota-
+tion close to the speed of light in the internal regions
+of an accretion disk, which yields a very strong differ-
+ence of Doppler shift effects on two sides of the black
+hole. This strong asymmetry of apparent luminosity
+is the main signature of the black hole image with a
+thin accretion disk. In short, the effects from Doppler
+shift and gravitational redshift drastically modify the
+luminosity distribution for the observed disk image at
+large distance.
+The black hole spin hardly affects the shape of the
+primary image.
+The principal effect of black hole
+spin is to change the radius of the marginally stable
+orbit and hence to modify the location of the inner
+edge of the accretion disk. Unlike in the case of a
+Schwarzschild black hole, a rapid rotation of Kerr
+black hole could lead to that the inner edge of the
+direct image coincides with the higher order images,
+so the dark gap between them may no longer exist
+[62, 63]. Due to the inner edge of the accretion disk
+being located far deeper in the gravitational poten-
+tial, the range of accessible redshift in the disk for the
+rapidly rotating Kerr black hole is far broader than
+for the Schwarzschild case. Thus, the higher order
+images round a rapidly spinning black hole carry less
+flux than in the Schwarzschild case, which means that
+they are much more difficult to spatially resolve from
+the direct image of the disk in the rapidly rotating
+black hole case.
+Moreover, the gravitational field of the accretion
+disk also affects the propagation of photon and fur-
+ther modifies the shape of black hole shadow. Re-
+cently, a static axially symmetric solution, which de-
+scribes the superposition of a Schwarzschild black
+hole with a relativistic thin and heavy accretion disk
+( Lemos-Letelier disk [64]), is applied to study black
+hole shadow [65]. This static disk with an inner edge
+is assumed to be made of two streams of counter-
+rotating particles [64], which leads to a total van-
+ishing angular momentum and ensures the existence
+of a static disk in equilibrium with the black hole.
+A heavy accretion disk yields some new features for
+the black hole image [65].
+There is a progressive
+optical enlargement of the disk image covering part
+of the shadow, despite the fact that the disk is in-
+finitesimally thin. This is a consequence of the in-
+creasing light rays’ bending towards the disk due to
+the increase of disk’s “weight”. The heavy disk also
+stretches the black hole shadow so that there is an ex-
+tra deformation of the shadow shape, which becomes
+more prolate as the disk contributes to a higher frac-
+tion of the total mass.
+Furthermore, the noninte-
+grability of the photon motion arising from a heavy
+accretion disk also leads to some chaotic patterns
+both in the black hole shadow and the disk image.
+These features also appear in the gravity system of
+a Schwarzschild black hole surrounded by a massive
+Bach-Weyl ring [37]. The chaotic lensing also leads
+to some distinct differences in the shape of photon
+sphere and the black hole shadow. This is because
+the chaotic orbits sharply modify the locally mea-
+sured four-momentum of the photons reaching a dis-
+tant observer and further influence the celestial coor-
+dinates of the images associated with these photons
+in the observer’s sky, and the latter directly deter-
+mines the shape of the black hole shadow and the
+disk image.
+IV.
+POLARIZED IMAGE OF A BLACK
+HOLE
+Electromagnetic wave is a kind of transverse waves
+so the optical image of a black hole must carry the
+polarization information about the light emitted from
+the accretion disk around the black hole. Recently,
+the EHT Collaboration has published the polarimet-
+ric image of the black hole M87∗ [13, 14]. The twist-
+ing polarization patterns revealed the existence of
+magnetic field near the black hole.
+It is the first
+time to measure the polarization information char-
+acterized by the magnetic field near the black hole,
+which is helpful to understand the formation of the
+black hole jet far from 55 million light years.
+Actually, in order to extract the information car-
+ried in the polarized image of a black hole, one must
+compare the observed polarimetry data with the the-
+oretical one. Thus, it is very vital to make theoretical
+analyses and numerical simulations on the polarized
+images for various black holes. In general, the po-
+larization structures in the black hole images depend
+on the details of the emitting plasma, principally the
+magnetic field geometry, and are also affected by the
+strongly curved spacetime near the black hole. For
+the origin of the polarized emission around a black
+hole, there is a typical scenario where the light with
+high polarization degrees, especially the linearly po-
+larized light, is produced by synchrotron emission in
+a compact and energetic region of the inner hot disk
+[66, 67]. It is because the relativistic Doppler beam-
+ing effect yields that the propagation directions of
+the photons emitted by a charged relativistic parti-
+cle are beamed almost along the tangent direction
+of the particle’s motion so that the light rays in the
+particle’s orbital plane are linearly polarized. In the
+cold disk model [57], the situation is different, the
+dominant thermal radiation leads to that the polar-
+ized directions of light waves are disorder so that the
+disk becomes a source of natural light without the
+total polarization. Thus, in the simulations of the
+polarized image of a black hole, only the hot disk
+model is considered. Moreover, as the linearly po-
+larized light passes through the outer magnetized re-
+gions in plasma, it further undergoes the Faraday
+depolarization effects [68–70].
+Along the path of each light ray from plasma to
+observer, the polarization components expressed by
+
+7
+the Stokes parameters (I, Q, U, V ) [71, 72] satisfy the
+polarized radiative transfer equations [73–77]
+dI
+dλ = J − KI,
+(5)
+where λ is an affine parameter. The Stokes vector
+I = g3(I, Q, U, V ), the propagation matrix K, and
+the emission vector J describe synchrotron emission
+and absorption coefficients in all Stokes parameters,
+as well as Faraday rotation and conversion. Thus,
+the propagations of the polarized light rays depend
+heavily on the plasma properties.
+In the general relativistic magnetohydrodynamic
+(GRMHD) simulations, the plasma in the hot disk
+around the supermassive black hole can be simplified
+by a model, where the plasma is assumed to be col-
+lisionless with electrons and ions so that the electron
+temperature Te deviates from the ion temperature Ti.
+The ratio between the temperatures Ti and Te can be
+expressed as [66, 67, 78]
+R = Ti
+Te
+= Rhigh
+β2
+1 + β2 + Rlow
+1
+1 + β2 ,
+(6)
+where β is the ratio of gas pressure to magnetic pres-
+sure. Rhigh and Rlow are numerical constants, which
+correspond to the ratio of ion to electron tempera-
+tures in the inner disk and in the jet region, respec-
+tively.
+Through quantitatively evaluating a large library
+of images based on GRMHD models and compar-
+ing with the resolved EHT 2017 linear polarization
+map of M87∗ [13, 14], the viable GRMHD models
+revealed that the characteristic parameters for aver-
+age intensity-weighted plasma in the emission region
+are the electron number density ne ∼ 104−5cm−3,
+the magnetic field strength B ≃ 7 − 30G, and the
+dimensionless electron temperature θe ∼ 8 − 60.
+Moreover, recent theoretical investigation shows
+that the polarization images of M87 jets are very
+sensitive to the black hole spin [66], which could pro-
+vide a new possibility for measuring the spin param-
+eter of a black hole. In the low-spin case, there are
+much more symmetric ring shape patterns. This is
+because the beaming and de-beaming effects are not
+so large and the jet acceleration is not so signifi-
+cant as the spin is small. In the high-spin case as
+a = 0.99MBH, the polarized image of the approach-
+ing jet disappeared in the low-spin case is clear [66].
+This is because the high black hole spin gives arise
+to that the particle motion in the plasma can be ac-
+celerated up to the Lorentz factor of ΓL ∼ 3 and
+further yields that the approaching jet is more bright
+than the counter one [66]. Furthermore, there is the
+crescent-like image produced by the toroidal motion
+of gas blobs, which demonstrates that the jet acceler-
+ation process strongly depends on the black hole spin
+[79].
+One can also extract information about circular
+polarization through analyzing the Stokes quantity
+V in the black hole images. The circular polarization
+can be amplified by the Faraday conversion in the
+well-ordered magnetic field.
+This is different from
+the case of the linear polarization where the polar-
+ization vectors are disordered by the strong Faraday
+rotation near the black hole. Generally, in a model
+with hot disk, the circular polarization light images
+are faint and turbulent because the hot region oc-
+cupied with chaotic magnetic fields is Faraday thick
+so that the Faraday conversion cannot be efficient.
+However, the study of circular polarization images
+is helpful to understand the polarized information in
+black hole images more completely.
+The combina-
+tion of linear and circular polarizations in future ob-
+servations could provide a higher-precision detection
+on the magnetic structure, the temperature distribu-
+tion and the coupling between proton and electron
+near black holes. It is shown that the circular polar-
+ization images are sensitive to the inclination angle
+[67]. Moreover, there is a “separatrix” in the circu-
+lar polarization images and across which the sign of
+the circular polarization is reversed. This can be at-
+tributed to the helical magnetic field structure in the
+disk [67]. It implies that future full polarization EHT
+images are quite useful tracers of the magnetic field
+structures near black holes.
+The numerical simulations for the polarization im-
+age of the black hole are generally computation-
+ally expensive due to the broad parameter surveys
+and the complicated couplings among astrophysical
+and relativistic effects. Recently, a simple model of
+an equatorial ring of magnetized fluid has been de-
+veloped to investigate the polarized images of syn-
+chrotron emission around the Schwarzschild black
+hole [80] and the Kerr black hole [81]. Although only
+the emission from a single radius is considered, this
+model can clearly reveal the dependence of the po-
+larization signatures on the magnetic field configu-
+ration, the black hole spin and the observer inclina-
+tion. Moreover, with this model, the image of a finite
+thin disk can be produced by simply summing con-
+tributions from individual radii. The studies [80, 81]
+also indicates that the ring model image is broadly
+consistent with the polarization morphology of the
+EHT image. However, one must note that this sim-
+ple ring model produces a high fractional polarization
+(≥ 60%) even after blurring, which is much larger
+than that in the M87∗ image where the resolved frac-
+tional polarization is about ≤ 20% [80]. This suggests
+that the significant depolarization from the internal
+Faraday effects is essential when modeling and in-
+terpreting the M87∗ image [82].
+Nevertheless, the
+success of the ring model in reproducing the struc-
+ture of some GRMHD images that have significant
+Faraday effects is encouraging for the prospects of
+physical inference from this simple model. Moreover,
+this simple model can be used to study the loops in
+the Stokes Q − U plane, which describes the con-
+tinuous variability in the polarization around a black
+
+8
+hole [83–88]. It is beneficial to understand some time-
+varying features of emission from a localized orbiting
+hotspot near black hole in the real astronomical envi-
+ronment. Thus, this model has been recently applied
+to study the polarized images of black holes in various
+spacetimes [89–93].
+In this simple ring model, the calculation of the po-
+larization vector usually resorts to a so-called Walker-
+Penrose quantity [94, 95]. It is conserved along the
+null geodesic in the spacetimes where the dynamical
+system of photon motion is integrable and the equa-
+tion of motion is full variable separable [94]. The con-
+served Walker-Penrose quantity builds a direct con-
+nection between the polarization vectors of photon
+starting from the emitting source and reaching the
+observer. So in such spacetimes, the propagation of
+polarization vectors can be calculated by analytical
+methods, which greatly simplifies the calculation of
+polarization vectors along null geodesics. However,
+in the spacetimes where the system of photon mo-
+tion is nonintegrable, such as, in the Bonnor black
+dihole spacetime [96], the Walker-Penrose quantity
+is no longer conserved along null geodesic. Without
+the help of the Walker-Penrose constant, the calcula-
+tion of the polarization vectors in this ring model may
+still rely on the numerical methods. In the Bonnor
+black dihole spacetime, there exist some fine fractal
+structures in the distribution of Stokes parameters Q
+and U in the polarized images [97]. The signs of Q
+and U are opposite for two adjacent indirect images.
+It could be caused by that the photons forming two
+adjacent indirect images are emitted from the up-
+per and lower surfaces of accretion disk, respectively,
+resulting in a large difference in the corresponding
+polarization vectors.
+V.
+APPLICATION PROSPECTS OF BLACK
+HOLE IMAGES
+The significance of studying black hole images lies
+in the following aspects. Firstly, such detections can
+identify black holes and further verify and test the
+theories of gravity including general relativity, and
+deepen our understandings on the nature of gravity.
+Secondly, analyzing information carried in black hole
+images enables us to understand matter distribution
+and physical processes around the black holes, and
+to give further insight into some fundamental prob-
+lems in physics. In the following, we present some
+potential application prospects of black hole images.
+A.
+Probe the matter distribution around black
+holes
+To probe the matter distribution around black
+holes, one must simulate images of black hole models
+by considering different choices and select a model
+that could accurately represent the main features of
+the observed images. For the black hole M87∗, it is
+well known that it belongs to the class of low luminos-
+ity active galactic nuclei, and its spectral energy dis-
+tribution presents features associated with emission
+from an optically thin and geometrically thick accre-
+tion disk ascribed to the synchrotron radiation with
+an observed brightness temperature in radio wave-
+lengths in the range of 109 − 1010K [1]. Recently,
+the most salient features appearing in the EHT Col-
+laboration images of M87∗ were reproduced with im-
+pressive fidelity and the corresponding configuration
+model revealed that there may exist an asymmet-
+ric bar-like structure attached to a two-temperature
+thin disk in the equatorial plane of the black hole
+[98]. Moreover, the asymmetry in brightness is a ro-
+bust indicator of the orientation of the spin axis. The
+simulations using different orientations of the black
+hole spin show that the spin direction opposite to
+the observed jet is favored by the asymmetric shape
+of the observed crescent sector.
+As mentioned in the previous part, the compari-
+son between the polarization patterns of the M87∗
+image and the viable GRMHD models reveals the
+existence of magnetic field near the black hole. Ac-
+tually, the magnetic field can generate some features
+of black hole images [99, 100]. For a rotating black
+hole immersed in a Melvin magnetic field [99], the
+shadow becomes oblate for the weak magnetic field.
+However, in the case with the strong magnetic field,
+the multiple disconnected shadows emerge, including
+a middle oblate shadow and many striped shadows.
+Moreover, the novel feature in the Melvin-Kerr black
+hole shadow is the gray regions on both sides of the
+middle main shadow [99], which are caused by the
+stable photon orbits around the stable light rings.
+In fact, the photons moving along the stable pho-
+ton orbits are trapped and they can’t enter the black
+hole. Strictly, the gray regions don’t belong to the
+black hole shadow, but if there are no light sources
+in the stable photon orbit regions, the observer also
+see dark shadows in the gray regions [99, 100]. The
+chaotic lensing arising from the magnetic field gives
+rise to the self-similar fractal structures in the black
+hole shadows. The chaotic image also occurs for the
+case illuminated by an accretion disk in the Kerr-
+Melvin black hole spacetime with a strong enough
+magnetic field [101]. These new effects in shadows
+could provide a new way to probe the magnetic field
+near black holes.
+The images of black holes indicate that the su-
+permassive black holes in the centers of galaxies are
+actually surrounded by plasma.
+Besides as a light
+source to illuminate black holes, plasma is a disper-
+sive medium where the index of refraction depends on
+the spacetime point, the plasma frequency and the
+photon frequency, so the plasma changes the path
+of the light traveling through it and further affects
+the geometrical features of black hole shadows [102–
+
+9
+119].
+The influence of plasma on the shadows de-
+pends mainly on the ratio between the plasma fre-
+quency and the photon frequency.
+If the plasma
+frequency is smaller than the photon frequency, the
+shadow is not very much different from the vacuum
+case. However, if the plasma frequency tends to the
+photon frequency, the significant changes in the pho-
+ton regions will lead to a drastic modification of the
+properties of the shadow. In the realistic case where
+the plasma frequency is much smaller than the pho-
+ton frequency, the plasma has a decreasing effect on
+the size of the shadows if the plasma density is higher
+at the photon sphere than at the observer position.
+The above analyses are based on an assumption of
+plasma with radial power-law density. Recent study
+of angular Gaussian distributed plasma [115], where
+the plasma is non-spherically symmetric, shows that
+the effect of plasma can be qualitatively explained by
+taking the plasma as a convex lens with the refractive
+index being less than 1. For the supermassive black
+holes at the centers of the Milky Way and the galaxy
+M87, which are the main targets of the current ob-
+servations by the EHT, it is shown that the plasma
+effects start to become relevant at radio wavelengths
+of a few centimeters or more. However, the present
+and planned instruments focus on the submillimeter
+range, where the scattering and self-absorption have
+no significant effect on the emitted radiation around
+the black holes and the plasma effects are very small
+[117, 118], so a realistic observation of the plasma in-
+fluence on the shadows seems unfeasible at present.
+B.
+Constrain black hole parameters and test
+theories of gravity
+It is natural to expect to constrain black hole pa-
+rameters by the using of shadows because the shape
+and size of shadows depend on the black hole parame-
+ters themselves. In general, since black hole shadows
+have complex shapes in the observer’s sky, the precise
+description of the shadow boundaries is crucial for
+measuring black hole parameters. To fit astronomical
+FIG. 6: The observables for the apparent shape of the
+Kerr black hole Rs and δs = Dcs/Rs [120].
+observations, several observables were constructed by
+using special points on the shadow boundaries in the
+celestial coordinates. For the Kerr black hole, the two
+observables Rs and δs = Dcs/Rs ( as shown in Fig.6)
+are introduced to measure the approximate size of the
+shadow and its deformation with respect to the refer-
+ence circle [120], respectively. If the inclination angle
+is given, the values of the mass and spin of the black
+hole can be obtained by the precise enough mea-
+surements of Rs and δs. Recently, the length of the
+shadow boundary and the local curvature radius are
+introduced to describe the shadow boundary [121].
+The black hole spin and the observer inclination can
+be constrained by simply measuring the maximum
+and minimum of the curvature radius.
+Moreover,
+a topological covariant quantity is analyzed to mea-
+sure and distinguish different topological structures
+of the shadows [122, 123]. To further describe the
+general characterization of the shadow boundaries, a
+coordinate-independent formalism [124] is proposed
+where the shadow curves Rψ(ψ) are expressed in
+terms of Legendre polynomials Rψ =
+∞
+�
+l=0
+clPl(cos ψ)
+with the expansion coefficients cl. The dimensionless
+deformation parameters δn are defined to measure
+the relative difference between the shadow at ψ = 0
+and at other angles ψ = π/n, n = 1, 2, ...k, and k is
+an arbitrarily positive integer. These distortions are
+both accurate and robust so they can also be imple-
+mented to analyse the noisy data.
+Above analyses are based on an assumption that
+the black hole shadows are cast by a bundle of pho-
+tons in parallel trajectories that originating at in-
+finity.
+For a realistic black hole surrounded by an
+accretion disk, the shadow is imprinted on the image
+of the accretion flow. In principle, comparing a de-
+tailed model of the accretion disk around the black
+hole with astronomical observations will yield a mea-
+surement of the size and shape of the shadow. How-
+ever, it is not feasible to predict the details of the
+brightness profile of the accretion flow image. The
+first reason is the incompleteness of accretion disk
+models, and all theoretical models are simplified by
+introducing some assumptions so they are impossi-
+ble to be completely consistent with the real disks.
+The other reason is the observed variability of the
+emission in the disk, since the inner accretion flow is
+highly turbulent and variable in the real astronomical
+environment. Thus, it is necessary to build a proce-
+dure to analyze the observation data that focuses on
+directly measuring the properties of the shadow in a
+manner that is not seriously affected by our inabil-
+ity to predict the brightness profile of the rest of the
+image [125]. The gradient method [126, 127] is such
+kind of model-independent algorithms in image pro-
+cessing, which has already been applied successfully
+to interferometric images to quantify the properties
+of the turbulent structure of the interstellar magnetic
+field. The basic concept in this algorithm is that the
+magnitude of the gradient of the accretion flow im-
+
+[M] 0
+-2
+0
+2
+-2
+D
+C2
+B
+0
+210
+age has local maxima at the locations of the steep-
+est gradients, such as, in the case of the expected
+EHT images, which coincide with the edge of the
+back hole shadow [125]. With the obtained gradient
+image where the rim of the black hole shadow appears
+as the most discernible feature, a shadow pattern al-
+gorithm matching with the Hough/Radon transform
+is employed to determine the shape and size of the
+shadow. This algorithm not only measures the prop-
+erties of the black hole shadow, but also assesses the
+statistical significance of the results.
+The distinct features of black holes originating
+from deviation parameters in the alternative theory
+can help test the general relativity. It is shown that
+the shadow becomes prolate for the negative devi-
+ation parameter and becomes oblate for the posi-
+tive one [128].
+The large deformation parameter
+in the Konoplya-Zhidenko rotating non-Kerr black
+hole yields the special cusp-shaped shadow for the
+equatorial observer [23]. The large deviation arising
+from the quadrupole mass moment leads to chaotic
+shadow and the eyeball-like shadows with the self-
+similar fractal structures [36]. The similar features
+of shadows also appear in other non-Einstein theories
+of gravity including the quadratic degenerate higher-
+order scalar-tensor theories [27]. Moreover, using the
+priori known estimates for the mass and distance of
+M87∗ based on stellar dynamics [1–6, 129–131], the
+inferred size of the shadow from the horizon-scale im-
+ages of the object M87∗ [1] is found to be consis-
+tent with that predicted from general relativity for a
+Schwarzschild black hole within 17% for a 68% con-
+fidence interval. However, this measurement still ad-
+mits other possibilities. The size of the black hole
+shadow M87∗ can be used as a proxy to measure
+the deviations from Kerr metric satisfied weak-field
+tests [132]. For the parameterized Johannsen-Psaltis
+black hole, it has four lowest-order parameters and
+the shadow depends primarily on the parameter α13
+and only weakly on spin [133]. The 2017 EHT mea-
+surement for M87∗ places a bound on the deviation
+parameter −3.6 < α13 < 5.9 [132]. For the modi-
+fied gravity bumpy Kerr metric [134], the size of the
+shadow depends primarily on the parameter γ1,2 and
+the requirement that the shadow size is consistent
+with the measurement of M87∗ within 17% gives a
+constraint on the deviation parameter −5.0 < γ1,2 <
+4.9 [132]. For the Konoplya-Rezzolla-Zhidenko met-
+ric [135], the EHT measurements results in the con-
+straint −1.2 < α1 < 1.3 [132]. For these parametric
+deviation metrics, the measurements of the shadow
+size lead to significant constraints on the deviation
+parameters that control the second post-Newtonian
+orders. This means that the EHT measurement of
+the size of a black hole leads to metric tests that
+are inaccessible in the weak-field tests. In general,
+such parametric tests cannot be connected directly
+to an underlying property of the alternative theory.
+Recently, the EHT measurements have been applied
+to set bounds on the physical parameters, such as,
+the electric charge [136] and the MOG parameter in
+the Scalar-Tensor-Vector-Gravity Theory [137]. The
+quality of the measurements [136] is already suffi-
+cient to rule out that M87∗ is a highly charged dila-
+ton black hole, a Reissner-Nordstr¨om naked singu-
+larity or a Janis-Newman-Winicour naked singularity
+with large scalar charge. Similarly, it also excludes
+considerable regions of the space of parameters for
+the doubly-charged dilaton and the Sen black holes.
+Such tests are very instructive [25, 138–140] because
+they can shed light on which underlying theories are
+promising candidates and which must be discarded
+or modified. The constraints and tests from shadows
+are complementary to those imposed by observations
+of gravitational waves from stellar-mass sources.
+Black hole shadow may also provide a way to test
+binary black hole. Nowadays, the gravitational-wave
+events detected by the LIGO-Virgo-KAGRA Collab-
+orations [141–145] confirm the existence of binary
+black hole system in the universe, and the systems
+of binary black hole are expected to be common as-
+trophysical systems.
+The shadows of the colliding
+between two black holes were simulated by adopt-
+ing the Kastor-Traschen cosmological multiblack hole
+solution, which describes the collision of maximally
+charged black holes with a positive cosmological con-
+stant [146, 147]. Fig.7 shows the change of the shad-
+ows with time t during the collision of the two black
+holes with equal mass. At t = 0, the two black holes
+are mutually away enough and their shadows are sep-
+arated. However, each shadow is a little bit elongated
+in the α direction because of the interaction between
+the two black holes. At t = 1.6, the eyebrowlike shad-
+ows appear around the main shadows. The eyebrow-
+like shadows can be explained by a fact that light rays
+bypass one black hole of binary system and enter the
+other one. With the further increase of time, the eye-
+browlike structures grow and the main shadows ap-
+proach each other. Although not discernible in the
+figure, in fact there appear the fractal structures of
+the eyebrows, i.e., infinitely many thinner eyebrows
+at the outer region of these eyebrows as well as at
+the inner region of the main shadows [147]. As time
+elapses, the interval between two black hole shadows
+becomes indefinitely narrower, and it is expected that
+the black hole shadows eventually merge with each
+other [147]. However, due to the special properties
+of Kastor-Traschen metric, the recent investigation
+also implies that there is no observer who will see the
+merge of black hole shadows even if the black holes
+coalesce into one [148]. Another important solution
+of binary black hole with analytical metric form is
+Majumdar-Papapetrou solution, which describes the
+geometry of two extremally charged black holes in
+static equilibrium where gravitational attraction is
+in balance with electrostatic repulsion.
+The simi-
+lar eyebrowlike shadows are found in the Majumdar-
+Papapetrou binary black hole system [149].
+
+11
+FIG. 7: The change of black hole shadows with time t
+during the collision of two equal mass black holes[147].
+Actually, these eyebrowlike shadows with fractal
+structures also appear in other binary black hole
+systems, such as, in the double-Schwarzschild and
+double-Kerr black hole systems [150] in which two
+black holes are separated by a conical singularity.
+These common key features imprinted in the shad-
+ows of binary systems, such as disconnected shadows
+with characteristic eyebrows, open up a new analytic
+avenue for exploring four dimensional black hole bi-
+naries [151].
+C.
+Fundamental problems in physics
+Dark matter The nature of dark matter is one
+of the most important open fundamental questions
+of physics.
+Dark matter is assumed to be an in-
+visible matter, which constitutes the dominant form
+of matter in the universe and has feeble couplings
+with the common visible matter at most.
+Despite
+extensive observational data supporting its presence
+on a large scale, dark matter has not been directly
+detected by any scientific instrument. Dark matter
+should influence black hole shadow due to its gravi-
+tational effects. A simple spherical model consisting
+of a Schwarzschild black hole with mass M and a
+homocentric spherical shell of dark matter halo with
+mass ∆M is applied to tentatively study the effects
+of dark matter on the black hole shadow [152]. It is
+found that the mass of dark matter and its distance
+over mass distribution lead to larger radius of shad-
+ows. However, it must be pointed out that in this
+simple model the dark matter is unlikely to manifest
+itself in the shadows of galactic black holes, unless
+its concentration near black holes is abnormally high
+[152].
+The effect of dark matter halo on black hole shad-
+ows has been studied in the spacetimes of a spher-
+ically symmetric black hole and of a rotating black
+hole [153–157]. It is shown that the structures of the
+black hole shadows in the cold dark matter (CDM)
+and scalar field dark matter (SFDM) halos are very
+similar to the cases of the Schwarzschild and Kerr
+black holes, respectively. Both dark matter models
+influence the shadows in a similar way and the sizes of
+the shadows increase with the dark matter parame-
+ter k ≡ ρcR3, where the characteristic density ρc and
+the radius R are related to the distribution of dark
+matter halo in two models. In general, the influence
+of the dark matter on the black hole shadows is mi-
+nor and only becomes significant when k increases to
+order of magnitude of 107 for both CDM and SFDM
+models [153]. The calculation of the angular radii of
+the shadows shows that the dark matter halo could
+influence the shadow of Sgr A∗ at a level of order of
+magnitude of 10−3µas and 10−5µas, for CDM and
+SFDM, respectively. However, it is out of the reach
+of the current astronomical instruments [153]. The
+current EHT resolution is ∼ 60µas at 230 GHz and
+will achieve 15µas by observing at a higher frequency
+of 345 GHz and adding more very long baseline inter-
+ferometry (VLBI) telescopes. The space-based VLBI
+RadioAstron [158] will be able to obtain a resolution
+of 1 − 10µas.
+This is still at least three orders of
+magnitude lower than the resolution required by the
+CDM model. The black hole shadow has been stud-
+ied for a rotating black hole solution surrounded by
+superfluid dark matter and baryonic matter. Using
+the current values for the parameters of the superfluid
+dark matter and baryonic density profiles for the Sgr
+A∗ black hole, it is shown that the effects of the super-
+fluid dark matter and baryonic matter on the sizes of
+shadows are almost negligible compared to the Kerr
+vacuum black hole [155]. Moreover, comparing with
+the dark matter, the shadow size increases consider-
+ably with the baryonic mass. This can be understood
+by the fact that the baryonic matter is mostly located
+in the galactic center. Similarly, the baryonic matter
+in this model yields an increase of the angular diam-
+eter of the shadow of the magnitude 10−5µas for the
+Sgr A∗ black hole [155].
+The axion is a hypothetical particle beyond the
+standard model, which is initially proposed to solve
+the strong CP (charge-conjugation and parity) prob-
+lem [159–162].
+Nowadays, axionlike particles are
+also introduced in fundamental theories and served
+as an excellent dark matter candidate so there are
+many search experiments designed to prob axions
+[163–168]. Axion cloud around a rotating black hole
+may be formed through the superradiance mecha-
+nism if the Compton wavelength of axion particle is
+at the same order of the black hole size [169, 170].
+Due to the existence of the axion cloud, the axion-
+electromagnetic-field coupling gives rise to that the
+position angles of linearly polarized photons emit-
+ted near the horizon oscillate periodically [171–174].
+Along this line, a novel strategy of detecting axion
+clouds around supermassive black holes is recently
+
+(M3)
+8-
+-4
+0
+4
+8
+8
+4
+0
+4
+8
+5
+0 (M3)o
+a. 1=
+.ar=}
+4
+-4
+-5
+0
+5
+S.C=t
+f=4'é
+4
+-4
+-5
+0
+5
+{=O [XJ\H]
+a.1=
+412
+FIG. 8: The expected axion parameter space probed by
+polarimetric observations of M87∗ (green) and Sgr A∗
+(red) for different position angle precisions [175].
+The
+bounds from CAST [167] (gray) and Supernova 1987A
+(pale yellow) are shown to make a comparison.
+proposed by using the high spatial resolution and
+polarimetric measurements of the EHT [175]. Fig.8
+presents the axion parameter space which is poten-
+tially probed by M87∗ and Sgr A∗ for different posi-
+tion angle precisions [175]. This method is comple-
+mentary to the constraints from the black hole spin
+measurements through gravitational wave detections
+[176].
+Since the position angle oscillation induced
+by the axion background does not depend on photon
+frequency, it is expected that polarimetric measure-
+ments at different frequencies in the future can be
+used to distinguish astrophysical background and to
+improve the sensitivity of tests of the axion superra-
+diance scenario. Moreover, the possibility of probing
+ultralight axions by the circular polarization light is
+also studied in [177].
+Extra dimension The possible existence of extra
+dimensions is one of the most remarkable predictions
+of the string theory.
+The extra spatial dimension
+could play an important role in fundamental theories
+within the context of the unification of the physical
+forces and also in black hole physics. For the high-
+dimensional black holes, it is shown that the extra
+dimension influences the shape and size of the shad-
+ows [151, 178–180]. Using the size and deviation from
+circularity of the shadow of the black hole M87∗ ob-
+served by the EHT collaboration, the curvature ra-
+dius of AdS5 in the Randall-Sundrum brane-world
+scenario is bounded by an upper limit l ≲ 170AU
+[181]. This upper limit is far from being competitive
+with current O (mm) scale constraints from preci-
+sion tests of gravity, but greatly improves the limit
+l ≲ 0.535 Mpc obtained from GW170817 [182]. More
+importantly, it is an independent limit from imaging
+the dark shadow of M87∗. Using a rotating black hole
+solution with a cosmological in the vacuum brane, the
+black hole shadow together with the observed data
+of M87∗ also provides a upper bound for the normal-
+ized tidal charge q < 0.004 [183], which is the second
+best result for the tidal charge to date and is a little
+higher than the best one q < 0.003 from a solar sys-
+tem test [184]. Moreover, the negative values of the
+tidal charge are reported to be favored with the M87∗
+and Sgr A∗ data in the brane contexts by the using
+of Reissner-Nordstr¨om-type geometry [185–187] and
+a rotating black hole without a cosmological constant
+[188].
+For the case of the compactified extra dimension,
+the shadow of a rotating uniform black string has
+been studied where the extra spatial dimension is
+treated as a compacted circle with the circumference
+l [189]. The momentum of photon arising from the
+fifth dimension enlarges the photon regions and the
+shadow of the rotating 5D black string while it has
+slight impact on the distortion. The angular diam-
+eter in the EHT observations of M87∗ leads to the
+constraint on the length of the compact extra di-
+mension 2.03125 mm ≲ l ≲ 2.6 mm [189].
+Simi-
+larly, from the observations of Sgr A∗, the constraints
+2.28070 mm ≲ l ≲ 2.6 mm and 2.13115 mm ≲ l ≲
+2.6 mm can be given by the upper bounds of the
+emission ring and the angular shadow diameter re-
+spectively [189]. In particular, within these bounds,
+the rotating 5D black string spacetime is free from
+the Gregory-Laflamme instability [189].
+Effects of the specific angular momentum ξψ of
+photon from the fifth dimension on black hole shadow
+have also been studied for a rotating squashed
+Kaluza-Klein black hole [190], which is a kind of
+interesting Kaluza-Klein type metrics with the spe-
+cial topology and asymptotical structure [191].
+It
+has squashed S3 horizons so the black hole has a
+structure similar to a five-dimensional black hole in
+the vicinity of horizon, but behaves as the four-
+dimensional black holes with a constant twisted S1
+fiber in the far region. For this special black hole, the
+radius Rs of the black hole image in the observer’s
+sky has different values for the photons with different
+angular momentum ξψ. The real radius of the black
+shadow is equal to the minimum value of Rsmin. Es-
+pecially, as the black hole parameters lie in a certain
+special range, it is found that there is no shadow
+for a black hole since the minimum value Rsmin = 0
+in these special cases [190], which is novel since it
+does not appear in the usual black hole spacetimes.
+It must be pointed out that the emergence of black
+hole without shadow does not mean that light rays
+can penetrate through the black hole. Actually, it is
+just because the photons near the black hole with cer-
+tain range of ξψ change their propagation directions
+and then become far away from the black hole. The
+phenomenon of black hole without black shadow will
+vanish if there exists the further constraint on the
+specific angular momentum ξψ of photon from the
+fifth dimension. In the case where black hole shadow
+exists, the radius of the black hole shadow increases
+monotonically with the increase of extra dimension
+
+[oa[w'(6N)]
+-SS
+-50
+-18
+-le
+4
+-5
+Q 0= 0'3。
+Q○=↓。
+roalc
+Q0=3。
+0
+Vo Tor:
+S
+313
+parameter in the non-rotating case.
+With the in-
+creasing of rotation parameter, the radius of the black
+hole shadow gradually becomes a monotonously de-
+creasing function of the extra dimension parameter.
+With the latest observation data, the angular radii
+of the shadows for the supermassive black hole Sgr
+A∗ at the centre of the Milky Way Galaxy and the
+supermassive black hole in M87 are estimated [190],
+which implies that there is a room for the theoreti-
+cal model of such a rotating squashed Kaluza-Klein
+black hole.
+Coupling between the photon and background field
+Analogous to the motion of charged particles in an
+electromagnetic field, the propagation of light rays
+in a spacetime is also influenced by the coupling be-
+tween the photon and background field, which could
+leave observable effects on the black hole shadow. In
+the standard Einstein-Maxwell theory, there is only a
+quadratic term of Maxwell tensor directly related to
+electromagnetic field, which can be seen as an inter-
+action between Maxwell field and metric tensor. Ac-
+tually, the interactions between electromagnetic field
+and curvature tensor could appear naturally in quan-
+tum electrodynamics with the photon effective ac-
+tion originating from one-loop vacuum polarization
+[192].
+Although these curvature tensor corrections
+appear firstly as an effective description of quantum
+effects, the extended theoretical models without the
+small coupling constant limit have been investigated
+for some physical motivations [193–196].
+The coupling between the photon and Weyl tensor
+leads to birefringence phenomenon so that the paths
+of light ray propagations are different for the cou-
+pled photons with different polarizations. Thus, it
+is natural to give rise to double shadows for a single
+black hole because the natural lights near the black
+hole can be separated into two kinds of linearly po-
+larized light beams with mutually perpendicular po-
+larizations [197]. With the increase of the coupling
+strength, the umbra of the black hole decreases and
+the penumbra increases. In the case of an equatorial
+thin accretion disk around the Schwarzschild black
+hole, the black hole image and its polarization dis-
+tribution are also affected by the coupling strength
+[198]. The observed polarized intensity in the bright
+region is stronger than that in the darker region. It
+is also noted that the effect of the coupling on the
+observed polarized vector is weak in general and the
+stronger effect appears in the bright region close to
+the black hole in the image plane. Moreover, for the
+different coupling strengths, the observed polarized
+patterns have a counterclockwise vortex-like distri-
+bution with a rotational symmetry as the observed
+inclination angle θ0 = 0◦.
+The rotational symme-
+try in polarized patterns gradually vanishes with the
+increase of the inclination angle. Quantum electro-
+dynamic effects from the Euler-Heisenberg effective
+Lagrangian on the shadow have been studied in the
+black hole background [199]. Similarly, in this case,
+the birefringence effect also yields that observer sees
+different shadow sizes of a single black hole for dif-
+ferent polarization lights.
+The coupling between a photon and a generic vec-
+tor field is also introduced to study black hole shadow
+[200].
+The generic vector field is assumed to obey
+the symmetries possessed by the black hole and the
+boundary condition that the vector field vanishes at
+infinity. It is found that the black hole shadow in
+edge-on view also has different appearances for differ-
+ent frequencies of the observed light. This is because
+the coupling form alters the way that the system de-
+pends on the initial conditions. These new phenom-
+ena about the black hole shadow originating from the
+coupling between the photon and background vector
+field are not simply caused by modifications of the
+metric, which could help give insight into new physics
+[200]. In particular, such a kind of coupling can affect
+the motion of photons and phenomenologically depict
+a violation of equivalence principle [200]. Thus, it is
+proposed as a mechanism to test the equivalence prin-
+ciple by analyzing black hole shadows. Although the
+current observation conditions might not allow us to
+directly detect these novel phenomena, it is expected
+that the future project of the next generation EHT
+with other future multi-band observations [201] as
+well as the related data-processing techniques could
+allow for tests of these new physics imprinted in the
+black hole shadows. Moreover, the shadow images
+of M87∗ and Sgr A∗ are recently used to constrain
+the parameters in the generalized uncertainty princi-
+ple (GUP) [202] and the Lorentz symmetry violation
+[203], respectively. Although these best upper limits
+are weaker than those obtained in most other physi-
+cal frameworks, they are valuable for further under-
+standing black hole images and fundamental prob-
+lems in physics [204–206].
+VI.
+SUMMARY
+The near-horizon images of the shadows of the su-
+permassive compact objects M87∗ and Sgr A∗ deliv-
+ered by the EHT have opened an amazing window
+for the strong-field test of gravity theories as well as
+fundamental physics. These images are composed of
+black hole shadow and the image of accretion disk
+around the central black hole. Black hole shadow is
+essentially formed by the light rays entering the black
+hole’s event horizon, in spite that its shape and size
+also depend on the position of observer and the types
+of light sources. The fundamental photon orbits and
+the invariant phase space structures determine the
+intrinsic features of the black hole shadow. However,
+the visualization of the shadow must resort to the
+emission in the accretion disk around the black hole
+in the real astronomical environment.
+This means
+that the visible images of the black hole also depend
+on the properties of the accretion disk and the phys-
+
+14
+ical processes in the disk, which yields that the black
+hole images could have a highly model-dependent ap-
+pearance [125]. For example, some models show a
+partially obscured shadow and others present an ap-
+parently exaggerated shadow. Especially, if the disk
+is optically thick, there may be no visible shadow
+at all, which means that the geometrical thickness is
+a key ingredient for observing the shadow. On the
+other hand, the information on luminance and po-
+larization stored in the image of accretion disk can
+be helpful to understand the matter distribution and
+structures in the strong field region near the black
+hole.
+Although black hole shadow and image carry the
+characteristic information of a black hole, it must be
+pointed out that the black hole shadows and images
+in some spacetimes may be not sensitive enough to
+certain parameters so that the effects of these param-
+eters on the black hole images can not be discrimi-
+nated in terms of the resolution of the current obser-
+vation devices. With the increasing accuracy and res-
+olution of the future astronomical observations and
+the technological development, as well as the more
+theoretical investigations, it is expected that these
+mint markings of black holes can be more clearly de-
+tected in the next generation EHT, the BlackHole-
+Cam and the space-based experiments. The future
+detections of the fractural fine structures in black
+hole shadows arising from the chaotic lensing and the
+competitive constraints on fundamental physics prin-
+ciples from black hole shadows will help better test
+theories of gravity and to deeply understand the fun-
+damental problems in modern physics. In a word, the
+study of black hole images is still in its infancy, and
+the detection of images for M87∗ and Sgr A∗ black
+holes is only a starting point.
+VII.
+ACKNOWLEDGMENTS
+We would like to thank Profs.
+Carlos Herdeiro
+and Jieci Wang for their useful comments and sug-
+gestions. This work was supported by the National
+Natural Science Foundation of China under Grant
+Nos. 12035005, 12275078 and 11875026.
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@@ -0,0 +1,1533 @@
+META TEMPORAL POINT PROCESSES
+Wonho Bae
+University of British Columbia & Borealis AI
+[email protected]
+Mohamed Osama Ahmed
+Borealis AI
+[email protected]
+Frederick Tung
+Borealis AI
+[email protected]
+Gabriel L. Oliveira
+Borealis AI
+[email protected]
+ABSTRACT
+A temporal point process (TPP) is a stochastic process where its realization is a
+sequence of discrete events in time. Recent work in TPPs model the process using
+a neural network in a supervised learning framework, where a training set is a
+collection of all the sequences. In this work, we propose to train TPPs in a meta
+learning framework, where each sequence is treated as a different task, via a novel
+framing of TPPs as neural processes (NPs). We introduce context sets to model
+TPPs as an instantiation of NPs. Motivated by attentive NP, we also introduce
+local history matching to help learn more informative features. We demonstrate
+the potential of the proposed method on popular public benchmark datasets and
+tasks, and compare with state-of-the-art TPP methods.
+1
+INTRODUCTION
+With the advancement of deep learning, there has been growing interest in modeling temporal point
+processes (TPPs) using neural networks. Although the community has developed many innovations
+in how neural TPPs encode the history of past events (Biloˇs et al., 2021) or how they decode these
+representations into predictions of the next event (Shchur et al., 2020; Lin et al., 2022), the general
+training framework for TPPs has been supervised learning where a model is trained on a collection
+of all the available sequences. However, supervised learning is susceptible to overfitting, especially
+in high noise environments, and generalization to new tasks can be challenging.
+In recent years, meta learning has emerged as an alternative to supervised learning as it aims to
+adapt or generalize well on new tasks, which resembles how humans can learn new skills from a
+few examples. Inspired by this, we propose to train TPPs in a meta learning framework. To this
+end, we treat each sequence as a “task”, since it is a realization of a stochastic process with its
+own characteristics. For instance, consider the pickup times of taxis in a city. The dynamics of
+these event sequences are governed by many factors such as location, weather and the routine of
+a taxi driver, which implies the pattern of each sequence can be significantly different from each
+other. Under the supervised learning framework, a trained model tends to capture the patterns seen
+in training sequences well, but it easily breaks on unseen patterns.
+As the goal of modeling TPPs is to estimate the true probability distribution of the next event time
+given the previous event times, we employ Neural Processes (NPs), a family of the model-based
+meta learning with stochasticity, to explain TPPs. In this work, we formulate neural TPPs as NPs
+by satisfying some conditions of NPs, and term it as Meta TPP. Inspired by the literature in NP, we
+further propose the Meta TPP with a cross-attention module, which we refer to as Attentive TPP. We
+demonstrate the strong potential of the proposed method through extensive experiments.
+Our contributions can be summarized as follows,
+• To the best of our knowledge, this is the first work that formulates the TPP problem in a
+meta learning framework, opening up a new research direction in neural TPPs.
+• Inspired by the NP literature, we present a conditional meta TPP formulation, followed by
+a latent path extension, culminating with our proposed Attentive TPP model.
+1
+arXiv:2301.12023v1  [cs.LG]  27 Jan 2023
+
+• The experimental results show that our proposed Attentive TPP model achieves state-of-
+the-art results on four widely used TPP benchmark datasets, and is more successful in
+capturing periodic patterns on three additional datasets compared to previous methods.
+• We demonstrate that our meta learning TPP approach can be more robust in practical de-
+ployment scenarios such as noisy sequences and distribution drift.
+2
+PRELIMINARIES
+Neural processes. A general form of optimization objective in supervised learning is defined as,
+θ∗ = arg max
+θ
+EB∼p(D)
+�
+� �
+(x,y)∈B
+log pθ(y | x)
+�
+�
+(1)
+where D := {(x(i), y(i))}|D|
+i=1 for an input x and label y, and B denotes a mini-batch set of (x, y)
+data pairs. Here, the goal is to learn a model f parameterized by θ that maps x to y as fθ : x → y.
+In recent years, meta learning has emerged as an alternative to supervised learning as it aims to
+adapt or generalize well on new tasks (Santoro et al., 2016), which resembles how humans learn
+new skills from few examples. In meta learning, we define a meta dataset, a set of different tasks,
+as M := {D(i)}|M|
+i=1 . Here, D(i) is a dataset of i-th task consisting of a context and target set as
+D := C ∪ T . The objective of meta learning is then defined as,
+θ∗ = arg max
+θ
+EBD∼p(M)
+�
+�
+�
+(C,T )∈BD
+log pθ(YT | XT , C)
+�
+�
+(2)
+where BD denotes a mini-batch set of tasks. Also, XT and YT represent inputs and labels of a target
+set, respectively. Unlike supervised learning, the goal is to learn a mapping from x to y given C:
+more formally, fθ(·, C) : x → y. Although meta learning is a powerful framework to learn fast
+adaption to new tasks, it does not provide uncertainty for its predictions, which is becoming more
+important in modern machine learning literature as a metric to measure the reliability of a model.
+To take the uncertainty into account for meta learning, Neural processes (NPs) have been proposed
+(Garnelo et al., 2018b;a). Instead of finding point estimators as done in regular meta learning models,
+NPs learn a probability distribution of a label y given an input x and context set C: pθ(y|x, C). In
+this work, we frame TPPs as meta learning instead of supervised learning, for the first time. To
+this end, we employ NPs to incorporate the stochastic nature of TPPs. In Section 3.1, we propose
+a simple modification of TPPs to connect them to NPs, which enables us to employ a rich line of
+works in NPs to TPPs as described in Section 3.2 and Section 3.3.
+Neural temporal point processes. TPPs are stochastic processes where their realizations are se-
+quences of discrete events in time. In notations, a collection of event time sequences is defined as
+D := {s(i)}|D|
+i=1 where s(i) = (τ (i)
+1 , τ (i)
+2 , . . . , τ (i)
+Li ) and Li denotes the length of i-th sequence. The
+history of studying TPPs started decades ago (Daley & Vere-Jones, 2003), but in this work, we focus
+on neural TPPs where TPPs are modeled using neural networks (Shchur et al., 2021). As described
+in Figure 1a, a general form of neural TPPs consists of an encoder, which takes a sequence of previ-
+ous event times and outputs a history embedding, and a decoder which takes the history embedding
+and outputs probability distribution of the time when the next event happens.
+Previous works of neural TPPs are auto-regressively modeled in a supervised learning framework.
+More formally, the objective of neural TPPs are defined as,
+θ∗ = arg max
+θ
+EB∼p(D)
+�
+�
+|B|
+�
+i=l
+Li−1
+�
+l=1
+log pθ(τ (i)
+l+1 | τ (i)
+≤l )
+�
+�
+(3)
+where B ∼ p(D) denotes a mini-batch of event time sequences. To frame TPPs as NPs, we need to
+define a target input and context set shown in Equation (2), from an event time history τ≤l, which
+will be described in the following section.
+2
+
+𝜏!
+Decoder
+Encoder
+(a) Neural TPP
+𝜏"#!
+𝜏$
+𝜏"
+…
+𝑟!
+𝑟$
+𝑟"
+…
+Attention
+Mask
+𝜏!
+Decoder
+Encoder
+(b) Conditional Meta TPP
+𝜏"#!
+𝜏$
+𝜏"
+…
+𝑟!
+𝑟$
+𝑟"
+…
+Attention
+Mask
+𝑟"%!
+𝐺
+𝜏!
+Decoder
+Encoder
+(c) Attentive TPP
+𝜏"#!
+𝜏$
+𝜏"
+…
+𝑟!
+𝑟$
+𝑟"
+…
+Attention
+Mask
+𝑟"%!
+𝐺
+𝜏!
+Encoder
+𝜏$
+𝜏"
+…
+𝑟!
+𝑟$
+𝑟"
+…
+𝑟"%!
+𝜇
+𝜎
+𝑧
+Latent Path
+Cross
+Attention
+𝑘!
+𝑟"
+&
+𝑘"#!
+𝑞
+…
+Weight 
+Sharing
+Figure 1: Overall architectures of TPP models.
+3
+META TEMPORAL POINT PROCESS AND ITS VARIANTS
+3.1
+TEMPORAL POINT PROCESSES AS NEURAL PROCESSES
+To frame TPPs as NPs, we treat each event time sequence s as a task for meta learning, which
+intuitively makes sense since each sequence is a realization of a stochastic process. For instance, the
+transaction times of different account holders are very different from each other due to many factors
+including an account holder’s financial status and characteristics.
+With the new definition of tasks, we define a target input and context set for a conditional probability
+distribution of meta learning shown in Equation (2), using previous event times τ≤l. There are many
+ways to define them but a target input and context set need to be semantically aligned since the target
+input will be an element of the context set for the next event time prediction. Hence, we define a
+target input for τl+1 as the latest “local history” τl−k+1:l where k is the window size of the local
+history. Similarly, a context set for τl+1 is defined as Cl := {τt−k+1:t}l−1
+t=1. Here, if t − k ≤ 0, we
+include event times from τ1. With Transformer structure, it is easy to efficiently compute the feature
+embeddings for the context set C. Figure 1b shows a schematic of the Conditional Meta TPP with a
+mask (shaded) used for an example case of 5 event times with a local history window size of k = 3.
+Then, the feature embedding rl contains information of τl−k+1:l. With the notations for target inputs
+and context sets, we propose the objective of TPPs in a meta learning framework as,
+θ∗ = arg max
+θ
+EB∼p(D)
+�
+�
+|B|
+�
+i=l
+Li−1
+�
+l=1
+log pθ(τ (i)
+l+1 | τ (i)
+l−k+1:l, C(i)
+l )
+�
+� .
+(4)
+Note that we have only one target label τ (i)
+l+1 to predict per event unlike the general meta learning
+objective in Equation (2) where usually |T | > 1. It is because TPP models in general are trained
+to predict the next event time. Modeling TPPs to predict multiple future event times would be an
+interesting future work, but it is out of scope of this work.
+Requirements for neural processes. Let XT := {xi}|T |
+i=1 and YT := {yi}|T |
+i=1 be a set of target
+inputs and labels, respectively, and π be an arbitrary permutation of a set. To design NP models, it
+is required to satisfy the following two conditions.
+Condition 3.1 (Consistency over a target set). A probability distribution pθ is consistent if it
+is consistent under permutation: pθ(YT | XT , C) = pθ(π(YT ) | π(XT ), C), and marginalization:
+pθ(y1:m | XT , C) =
+�
+pθ(y1:n | XT , C) dym+1:n for any positive integer m < n.
+3
+
+Condition 3.2 (Permutation invariance over a context set). pθ(YT | XT , C) = pθ(YT | XT , π(C))
+According to Kolmogorov extension theorem (Oksendal, 2013), a collection of finite-dimensional
+distributions is defined as a stochastic process if condition 3.1 is satisfied. In NP literature, condition
+3.1 is satisfied through factorization: it assumes target labels are independent to each other given a
+target input and a context set C, in other words, pθ(YT | XT , C) = Π|T |
+i=1pθ(yi | xi, x<i, y<i, C) ≈
+Π|T |
+i=1pθ(yi | xi, C) (Dubois et al., 2020). This assumption can be unrealistic if target labels are
+strongly dependent to previous target inputs and labels even after context representations are ob-
+served. It is, however, not necessary to assume factorization to make TPPs as NPs. As previously
+mentioned, we only care about predicting the next event time, which means |YT | = 1. When a
+set contains only one element, its permutation is always itself. More formally, the consistency un-
+der permutation of Condition 3.1 in TPPs: pθ(τl+1 | τl−k+1:l, Cl) = pθ(π(τl+1) | π(τl−k+1:l), Cl),
+is satisfied since π({τl+1}) = {τl+1} and π({τl−k+1:l}) = {τl−k+1:l}.
+Also, the marginal-
+ization under permutation in Condition 3.1 is satisfied as marginalization is not applicable for
+pθ(τl+1 | τl−k+1:l, Cl) since the target label set τl+1 contains only one element.
+Recall that NP models learn a probability distribution of a target label pθ given a target input and
+context set. For computational efficiency (to make inference O(|C| + |T |) time), the feature repre-
+sentation of C should be invariant to the size of the context set, for which Condition 3.2 is required.
+To satisfy Condition 3.2, we average-pool all the context features r1, r2, . . . rl−1 to generate the
+global feature for a task G as shown in Figure 1b, and term it as conditional Meta TPP following
+the terminology used in the NP literature. Each context feature r1, r2, · · · , rl−1 represents a feature
+from a transformer encoder such as Transformer Hawkes Processes (THP), that encodes the corre-
+sponding local history of the context set Cl. For instance, ri contains information of τi−k+1:i. To
+make ri only encode the subset of previous event times τi−k+1:i (instead of the whole previous event
+times τ≤i), we mask out events that are outside of the local history window using an attention mask
+as shown in Figure 1(b) and (c), which is different from a regular attention mask shown in Figure
+1(a). Using the permutation invariant feature G not only satisfies Condition 3.2, but also lets the de-
+coder approximate the probability distribution of a target label given both a target input and context
+set instead of just a target input. Now that we satisfy both requirements with a new architectural
+design, we can treat TPPs as NPs.
+Implementation. It can be expensive to compute the individual context feature rt for all 1 ≤ t < l,
+from each element of the context set τt−k+1:t ∈ Cl: the time complexity of computing all the
+context features for a sequence is O(L2). Instead of passing each element of a context set, using the
+Transformer architecture (Vaswani et al., 2017), we can simply pass the event times to obtain all the
+context features at once, of which time complexity is O(kL) where k is the window size of a local
+history. To this end, we employ the THP as the encoder. Please refer to Zuo et al. (2020) for details.
+3.2
+META TEMPORAL POINT PROCESS
+In the NP literature, NPs are generally modeled as latent variable models. Instead of using the
+deterministic global feature G as an input to the decoder (Garnelo et al., 2018a), a latent variable z is
+sampled from a probability distribution e.g. multi-variate Gaussian, using parameters inferred from
+the global feature G (Garnelo et al., 2018b). As it is intractable to compute the log-likelihood for
+a latent variable model, amortized variational inference (VI) can be used to approximate inference.
+In the setting of TPPs, the evidence lower bound (ELBO) of variational inference with an inference
+network pθ(z | CL) can be derived as,
+log pθ(τl+1 | τl−k+1:l, Cl) = log
+�
+pθ(τl | τl−k+1:l, z)pθ(z | Cl)dz
+(5)
+≥ Ez [log pθ(τl+1 | τl−k+1:l, z)] − KL(pθ(z | CL) | pθ(z | Cl))
+(6)
+≈ 1
+N
+N
+�
+n=1
+log pθ(τl | τl−k+1:l, zn) − KL(pθ(z | CL) | pθ(z | Cl))
+(7)
+where N denotes the number of samples of z ∼ pθ(z | CL) for Monte-Carlo approximation. Here,
+pθ(z | CL) is the posterior given the context at the last (L-th) event, which contains all the events
+of the sequence s (it is accessible in training time). Minimizing KL-divergence between pθ(z | CL)
+and pθ(z | Cl) is to make the global latent variable z inferred from Cl to be similar to the latent
+4
+
+variable of a sequence z from CL, in training time. To sample z, we use the reparameterization trick
+as z = µ + σ ⊙ ϵ where ϵ ∼ N(0, I) as described in the latent path of Figure 1c. In inference,
+we approximate evaluation metrics such as negative log-likelihood or root mean squared error using
+Monte-Carlo samples. But, as we do not have access to CL at l-th event when l < L, we use z from
+pθ(z | Cl). The detailed description of the evaluation metrics are provided in Appendix C.
+An advantage of a latent variable model is that it captures stochasticity of functions, which can
+be particularly beneficial to model TPPs since TPPs are stochastic processes. Experiments in Sec-
+tion 5.4 demonstrate it indeed helps to model TPPs over the deterministic case. In particular, it is
+robust to noises (Section 5.2). We term the latent variable model as Meta TPP throughout the paper.
+Discussion. The existing TPP models treat all event sequences as realization of the same process
+whereas the Meta TPP treats each sequence as a realization of a distinct stochastic process. We
+achieve this by conditioning on the global latent feature z that captures task-specific characteristics.
+For z to be task-specific, it has to be distinct for different sequences but similar throughout different
+events l ∈ [1, L − 1] within the same sequence. It is natural for the global features to be distinct by
+sequence but we need further guidance to make the global feature shared across all the event times in
+a sequence. Due to the permutation invariance constraint implemented in average-pooling, z cannot
+be very different at different event time: adding some addition context feature ri will not change G
+as well as z much. In addition, the KL-divergence between pθ(z | CL) and pθ(z | Cl) further enhances
+the task-specific characteristics of z. We provide more detailed discussion in Appendix B
+3.3
+ATTENTIVE TEMPORAL POINT PROCESS
+Early works in NPs suffered from the underfitting problem. To alleviate this, Kim et al. (2019)
+proposed AttentiveNP, which explicitly attends the elements in a context set to obtain a better fea-
+ture for target inputs. Inspired by this, we add a cross-attention module that considers the sim-
+ilarity between the feature of a target input and previous event times as described in Figure 1c.
+Given the local history (context) features r1, r2, . . . rl−1 at l-th time step, the key-query-value pairs
+K ∈ Rl−1×D, q ∈ R1×D, and V ∈ Rl−1×D for the cross-attention, are computed using their
+corresponding projection weights WK ∈ RD×D, WQ ∈ RD×D as,
+K = R · WK, q = rl
+T · WQ, V = R where R = [r1, r2, . . . , rl−1]T .
+(8)
+Here, K corresponds to [k1, k2, . . . , kl−1]T in Figure 1c. The feature of i-th attention head hi are
+then computed as follows,
+hi = Softmax(q · KT /
+√
+D) · V.
+(9)
+With W ∈ RHD×D and some fully connected layers denoted as FC, r′
+l ∈ R1×D is computed as,
+r′
+l = FC( [h1, h2, . . . , hH] · W ).
+(10)
+Finally, the decoder takes the concatenated feature of z, rl, and r′
+l as an input to infer a distribution.
+In the TPP setting, it is common that there are multiple periodic patterns in the underlying stochastic
+process. The cross-attention module provides an inductive bias to a model that the repeating event
+subsequences should have similar features. Our experiments in Section 5.2 demonstrate that the
+explicit attention helps to model TPPs in general, especially when there are periodic patterns.
+Decoder. The decoder takes the concatenated feature of the global latent feature z, target input
+feature rl that encodes τl−k:l−1, and attention feature r′ from the attention module. For the Meta
+TPP (without the attention module), the decoder takes as input the concatenated feature of z and
+rl. Here, z, rl, and r′ are all D-dimensional vectors. The decoder consists of two fully connected
+layers, and the input and hidden dimension of the decoder layers are either 2D or 3D depending on
+whether we use the feature from the attention module r′.
+The decoder outputs the parameters of the probability distribution of the next event time or
+pθ(τl+1 | τl−k+1:l, zm). Inspired by the intensity-free TPP (Shchur et al., 2020), we use a mix-
+ture of log-normal distributions to model the probability distribution. Formally, for l ∈ [1, L − 1],
+τl+1 ∼ MixLogNorm(µl+1, σl+1, ωl+1) where µl+1 are the mixture means, σl+1 are the stan-
+dard deviations, and ωl+1 are the mixture weights.
+5
+
+4
+RELATED WORK
+Neural temporal point processes. Neural temporal point processes (NTPPs) have been proposed
+to capture complex dynamics of stochastic processes in time. They are derived from traditional
+temporal point processes (Hawkes, 1971; Isham & Westcott, 1979; Daley & Vere-Jones, 2003).
+Models based on RNNs are proposed by (Du et al., 2016) and (Mei & Eisner, 2017) to improve
+NTPPs by constructing continuous-time RNNs. More recent works use Transformers to capture
+long-term dependency (Kumar et al., 2019; Zhang et al., 2020; Zuo et al., 2020; Yang et al., 2022;
+Zhang et al., 2022). (Omi et al., 2019; Shchur et al., 2020; Sharma et al., 2021) propose intensity-free
+NTPPs to directly model the conditional distribution of event times.
+Omi et al. (2019) propose to model a cumulative intensity with a neural network. But, it suffers from
+problems that the probability density is not normalised and negative event times receives non-zero
+probabilities. Alternatively, Shchur et al. (2020) suggest modelling conditional probability density
+by log-normal mixtures. Transformer-based models like Zuo et al. (2020); Zhang et al. (2020)
+propose to leverages the self-attention mechanism to capture long-term dependencies. Another class
+of TPP methods called Neural Flows (Biloˇs et al., 2021), are proposed to model temporal dynamics
+with ordinary differential equations learned by neural networks. Unlike the previous TPP methods,
+we frame TPPs as meta learning (not supervised learning) for the first time.
+Neural processes. Meta learning is a learning framework that aims to adapt or generalize well on
+new tasks. There are three approaches in meta learning: metric-based (Koch et al., 2015; Vinyals
+et al., 2016; Sung et al., 2018; Snell et al., 2017), model-based (Santoro et al., 2016; Munkhdalai &
+Yu, 2017; Grant et al., 2018) and optimization-based (Finn et al., 2017; 2018; Nichol et al., 2018).
+Neural processes (NPs) is the model-based meta learning with stochasticity. Garnelo et al. (2018a)
+propose a conditional neural process as a new formulation to approximate a stochastic process us-
+ing neural network architecture. It succeeds the advantage of Gaussian Processes (GPs) as it can
+estimate the uncertainty of its predictions, without having expensive inference time. Garnelo et al.
+(2018b) generalize a conditional neural process by adding latent variables, which are approximated
+using variational inference. Although NPs can adapt to new tasks quickly without requiring much
+computation, it suffers from underfitting problem. To alleviate it, Kim et al. (2019) propose a cross-
+attention module, which explicitly attends the elements in the context set to obtain better representa-
+tions for the elements in the target set. As another way to address the underfitting problem, Gordon
+et al. (2020) propose a set convolutional layer under the assumption of translation equivariance of
+inputs and outputs, which is expanded to the latent variable counterpart in Foong et al. (2020).
+Transformer NP (Nguyen & Grover, 2022) is the most relevant work to ours. Although it also models
+event sequences, it focuses on modeling regular time series: discrete and regularly-spaced time
+inputs with corresponding label values. TPPs are different as they are continuous and irregularly-
+spaced time sequences not necessarily with corresponding label values.
+5
+EXPERIMENTS
+5.1
+EXPERIMENT SETTING
+Datasets. To compare the effectiveness of models, we conduct experiments on 4 popular benchmark
+datasets – Stack Overflow, Mooc, Reddit, and Wiki, and 3 datasets with strong periodic patterns we
+introduce – Sinusoidal wave, Uber, and NYC Taxi. Please refer to Appendix H for details.
+Metrics. We use the root mean squared error (RMSE) as the main metric along with the negative
+log-likelihood (NLL) as a reference since NLL can go arbitrary low if probability density is placed
+mostly on the ground truth event time. RMSE may not be a good metric, either, if one ignores
+stochastic components of TPPs and directly trains a baseline on the ground truth event times to
+obtain point estimations of event times (Shchur et al., 2021). We train all the methods on NLL
+and obtain RMSE in test time to not abuse RMSE scores, keeping stochastic components of TPPs.
+For marked TPP datasets, we extend the proposed method to make class predictions, and report
+accuracy. For details about marked cases, please refer to Appendix G.
+Baselines. We use intensity-free TPP (Shchur et al., 2020), Neural flow (Biloˇs et al., 2021), and
+Transformer Hawkes Processes (THP) (Zuo et al., 2020) as baselines. For intensity-free TPP and
+6
+
+Methods
+Stack Overflow
+Mooc
+Reddit
+Wiki
+RMSE
+NLL
+Acc
+RMSE
+NLL
+Acc
+RMSE
+NLL
+Acc
+RMSE
+NLL
+Acc
+Intensity-free
+3.64
+3.66
+0.43
+0.31
+0.94
+0.40
+0.18
+1.09
+0.60
+0.60
+7.76
+0.26
+(0.26)
+(0.02)
+(0.005)
+(0.006)
+(0.03)
+(0.004)
+(0.006)
+(0.04)
+(0.008)
+(0.05)
+(0.40)
+(0.03)
+Neural flow
+–
+–
+–
+0.47
+0.43
+0.30
+0.32
+1.30
+0.60
+0.56
+11.55
+0.05
+–
+–
+–
+(0.006)
+(0.02)
+(0.04)
+(0.04)
+(0.33)
+(0.07)
+(0.05)
+(2.22)
+(0.01)
+THP+
+1.68
+3.28
+0.46
+0.18
+0.13
+0.38
+0.26
+1.20
+0.60
+0.17
+6.25
+0.23
+(0.16)
+(0.02)
+(0.004)
+(0.005)
+(0.02)
+(0.004)
+(0.005)
+(0.04)
+(0.007)
+(0.02)
+(0.39)
+(0.03)
+Attentive TPP
+1.15
+2.64
+0.46
+0.16
+-0.72
+0.36
+0.11
+0.03
+0.60
+0.15
+6.25
+0.25
+(0.02)
+(0.02)
+(0.004)
+(0.004)
+(0.02)
+(0.003)
+(0.002)
+(0.04)
+(0.007)
+(0.01)
+(0.38)
+(0.03)
+Table 1: Comparison of the Attentive TPP to the state-of-the-art methods on a bootstrapped test sets.
+Methods
+Sinusoidal
+Uber
+NYC Taxi
+RMSE
+NLL
+RMSE
+NLL
+RMSE
+NLL
+Intensity-free
+1.29 (0.08)
+0.88 (0.02)
+51.23 (2.89)
+4.46 (0.02)
+46.59 (26.16)
+2.06 (0.07)
+Neural flow
+1.13 (0.07)
+0.99 (0.02)
+–
+–
+–
+–
+THP+
+1.72 (0.10)
+0.84 (0.02)
+90.25 (4.53)
+3.63 (0.03)
+10.31 (0.47)
+2.00 (0.01)
+Attentive TPP (Ours)
+1.45 (0.11)
+0.66 (0.02)
+22.11 (1.94)
+2.89 (0.04)
+8.92 (0.42)
+2.00 (0.009)
+Table 2: Experiment results on bootstrapped test sets with strong periodic patterns.
+neural flow, we add the survival time of the last event to NLL and fix some bugs specified in their
+public repositories. THP and its variants are originally based on intensity: they predict intensities
+from which log-likelihood and expectation of event times are computed. It is, however, computa-
+tionally expensive to compute them as it requires to compute integrals: especially, to compute the
+expected event times, it requires to compute double integrals, which can be quite expensive and
+complex to compute even with thinning algorithms described in Mei & Eisner (2017). To work
+around it without losing performance, we add the mixture of log-normal distribution proposed in
+(Shchur et al., 2020) as the decoder, and we call it THP+. For fair comparison, we fix the number
+of parameters of the models in between 50K and 60K except the last fully-connected layer for class
+predictions since it depends on the number of classes.
+Hyperparameters. We grid-search on every combination of dataset and method for learning rate
+∈ {0.01, 0.001, 0.0001, 0.00001} and weight decay ∈ {0.01, 0.001, 0.0001, 0.00001} for fair com-
+parison. We bootstrap for 200 times on test sets to obtain the mean and standard deviation (in
+parentheses) for the metrics in Figure 2a and Table 1–3 following Yang et al. (2022). All the other
+hyperparameters are fixed throughout the experiments, and are reported in Appendix I.
+5.2
+EXPERIMENT RESULTS
+In this section, we begin by comparing our attentive meta temporal point process (denoted as Atten-
+tive TPP) with state-of-the-art supervised TPP methods on 4 popular benchmarks. We then inves-
+tigate how Attentive TPP captures periodic patterns, and show how Attentive TPP can be used to
+impute missing events in noisy sequences. Finally, we consider robustness under distribution drift.
+Comparison with state-of-the-art methods. Table 1 summarizes our comparison of Attentive TPP
+with state-of-the-art baselines – intensity-free (Shchur et al., 2020), neural flow (Biloˇs et al., 2021)1,
+and THP+ (Zuo et al., 2020) on the Stack Overflow, Mooc, Reddit, and Wiki benchmarks. THP+
+generally performs better than the intensity-free and neural flow baselines. Attentive TPP further
+improves over THP+ on all datasets and metrics except for mark accuracy on Mooc and Wiki.
+Periodic patterns. As previously mentioned in Section 3.3, the cross-attention module is designed
+to capture periodic patterns by matching the local history of the current event to the local histories
+1Neural flow results on Uber, NYC Taxi and Stack Overflow (in Table 1) datasets are missing because the
+official implementation runs into NaN values for long event sequences in inversion step.
+7
+
+10
+20
+30
+40
+50
+60
+70
+Drop ratio (%)
+0
+10
+20
+30
+40
+50
+RMSE
+THP+
+Meta TPP
+Attn TPP
+(a) Imputation with different drop rates
+Jan-Feb Mar-AprMay-Jun Jul-Aug Sep-Oct Nov-Dec
+Target domains
+20
+40
+60
+80
+100
+120
+140
+RMSE
+THP+
+Meta TPP
+(b) Distribution drift in NYC Taxi dataset
+Figure 2: Experiment results on imputation and distribution drift.
+Attention
+Latent
+Reddit
+Uber
+RMSE
+NLL
+Acc
+RMSE
+NLL
+
+
+0.16
+0.92
+0.59
+63.71
+3.68
+
+
+0.13
+-0.39
+0.61
+63.35
+3.25
+
+
+0.12
+0.29
+0.61
+47.91
+3.72
+
+
+0.12
+0.07
+0.60
+21.87
+2.98
+Table 3: Comparison of the variants of Meta TPPs
+Reddit
+Methods
+# Params
+RMSE
+NLL
+Acc
+THP+
+113K
+0.26
+1.19
+0.60
+170K
+0.29
+0.79
+0.59
+226K
+0.28
+1.44
+0.57
+AttnTPP
+222K
+0.12
+0.07
+0.60
+Table 4: Comparison of diff. model size
+of the previous event times, in addition to alleviating the underfitting problem. To validate the effec-
+tiveness of the cross-attention, we experiment on the datasets with strong periodicity – Sinusoidal,
+Uber, and NYC Taxi (please refer to Appendix H for details). Table 2 shows that the Attentive TPP
+generally outperforms the state-of-the-art methods, except for RMSE on Sinusoidal. To investigate
+the behavior of the cross-attention, we provide an example in Figure 3a where we highlight 15 (out
+of 64) the most attended local history indices (in red) to predict the target event (in green) in a se-
+quence from Sinusoidal. The dotted grey lines represent the start and end of periods. We can observe
+that the attention refers to the local histories with similar patterns more than the recent ones.
+5.3
+APPLICATIONS
+Imputation. We study the robustness of Meta and Attentive TPP to noise by randomly dropping
+events, simulating partial observability in a noisy environment, and measuring imputation perfor-
+mance. For the experiment, we drop n percentage of all the event times drawn independently at
+random per sequence on the Sinusoidal wave dataset. In Figure 2a, we report the bootstrapped im-
+putation performance of THP+, Meta TPP, and Attentive TPP, in terms of RMSE. As the drop ratio
+increases, RMSE increases for all three models but the gap exponentially increases. Given that the
+performance gap between three models on ‘next event’ predictions is not as large (mean RMSE –
+THP+: 1.72, Meta TPP: 1.49, Attentive TPP: 1.45), the results shown in Figure 2a imply that the
+Meta and Attentive TPP are significantly more robust to the noise coming from partial observability.
+Distribution drift. Distribution drift occurs when the distribution observed during training becomes
+misaligned with the distribution during deployment due to changes in the underlying patterns over
+time. This is a common deployment challenge in real-world systems. Figure 2b shows how THP+
+and Meta TPP models trained on the January-February data of the NYC Taxi dataset generalize to
+subsequent months. Both models show a decrease in performance, suggesting the presence of non-
+stationary or seasonal patterns in the data that are not captured in the training months; however, Meta
+TPP is comparatively more robust across all out-of-domain settings. It is also worth mentioning that
+although the Attentive TPP generally performs better than Meta TPP in the conventional experimen-
+tal setting, it is not the case for distribution drifts. We conjecture it is because the cross-attention is
+designed to alleviate the underfitting problem, which results in being less robust to distribution drift.
+8
+
+0
+10
+20
+30
+40
+50
+Unit
+0
+1
+2
+3
+4
+5
+Bin count
+Most attended events
+Target event
+(a) Self-attention in THP and proposed cross-attention
+0
+10
+20
+30
+Bin counts
+Attentive TPP
+Targets
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+Days
+0
+10
+20
+30
+Bin counts
+THP+
+Targets
+(b) Visualization of test predictions vs. targets
+Figure 3: Qualitative analysis on the cross-attention and prediction results.
+5.4
+ABLATION STUDIES
+Meta TPP and its variants. In Table 3, we compare the proposed Meta TPP and its variants
+on Reddit, and Uber datasets. The result shows that both cross-attention and latent variable path
+generally help to improve the performance. When they are combined (resulting in the Attentive
+TPP), it generally performs the best in terms of both RMSE and NLL.
+Different model sizes. Both latent path and cross-attention components introduce additional learn-
+able parameters (for the Reddit dataset with 984 classes, THP+: 113K, Meta TPP: 126K, and Atten-
+tive TPP: 209K parameters). We provide an ablation study with varying number of model parameters
+for the THP+ baseline to validate the performance improvement does not come from the increased
+number of model parameters. In Table 4, we increase the number of model parameters for THP+ on
+Reddit along with the result of the Attentive TPP. The result shows that the larger model does not
+necessarily help to improve the performance: as the number of parameters increases, NLL some-
+times improves but it may hurt RMSE as in the case of Table 4. The significant improvement in
+performance of our proposed method shows the importance of providing an effective inductive bias.
+Parameter sharing. In Attentive NP, the encoders of the latent path and attention path are separated
+to provide different features. However, it can significantly increase computational overhead, and for
+this reason, we share the weights for the encoders. As an ablation study, we provide the performance
+with and without sharing the weights for the encoders of the Attentive TPP on the Stack Overflow
+dataset. Although the number of parameters of ‘with sharing’ is 50% less than ‘without sharing’
+(‘without sharing’: 86K vs. ‘with sharing’: 136K), it performs better than ‘without sharing’ (RMSE
+/ NLL – ‘without sharing’: 1.08 / 3.20 vs. ‘with sharing’: 1.03 / 2.81).
+Visualization of event time predictions. In TPP literature, the evaluation relies only on the RMSE
+and NLL metrics. It is, however, often hard to measure how practically useful a trained TPP model is.
+To qualitatively evaluate TPP models, we convert an event time sequence into time series sequence:
+we count the number of event times falling into each bin (in Figure 3b, each bin is a day). Figure 3b
+shows how close overall predictions of the Attentive TPP and THP+ (in red) are to the ground truth
+event times (in blue). In the figure, we can see that the Attentive TPP’s predictions closely align
+with the targets whereas the predictions of the THP+ are off at some regions. Note that as the y-axis
+represents bin counts, even a slight off from the ground truth implies large values in terms of RMSE.
+6
+CONCLUSION
+Previously, neural temporal point processes (TPPs) train neural networks in a supervised learning
+framework. Although performing well on event sequences similar to a training set, they are sus-
+ceptible to overfitting, and may not generalize well on sequences with unseen patterns. To alleviate
+this, we proposed a novel framework for TPPs using neural processes (NPs). We proposed the Meta
+TPP where TPP is formulated as NP, and further developed the Attentive TPP using a cross-attention
+module, which forces a model to use similar features for repeating local history patterns. Our exper-
+iments demonstrate that the proposed models outperform strong state-of-the-art baselines on several
+event sequence datasets, effectively capture periodic patterns, and increase robustness to noise and
+distribution drift. We believe this work opens up a new research direction to meta learning for TPPs.
+9
+
+REPRODUCIBILITY STATEMENT
+Hyperparameters and implementation details are available in Section 5.1 and Appendix I. The code
+for the baselines and the proposed method will be released upon acceptance. Our code is based on
+publicly available official intensity-free and neural flow code.
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+11
+
+A
+DERIVATION OF THE EVIDENCE LOWER BOUND
+We provide the detailed steps to derive the evidence lower bound of the proposed Meta TPP model
+shown in Equation (7). Here, Equation (8) to (9) holds with Jensen’s inequality.
+log pθ(τl | τl−k:l−1, Cl) = log
+�
+pθ(τl | τl−k:l−1, z)pθ(z | Cl)dz
+= log
+�
+pθ(z|CL) pθ(z|Cl)
+pθ(z|CL)pθ(τl|τl−k:l−1, z)dz
+≥
+�
+pθ(z|CL) log pθ(z|Cl)
+pθ(z|CL)pθ(τl|τl−k:l−1, z)dz
+= Ez∼pθ(z | CL) [log pθ(τl | τl−k:l−1, z)] − KL(pθ(z | CL) | pθ(z | Cl)).
+B
+ROLE OF GLOBAL LATENT FEATURE
+For the global latent feature z to be task-specific, it has to be distinct for different sequences but
+similar throughout different event times l ∈ [1, L − 1] within the same sequence as mentioned in
+Section 3.2. It is natural for the global features to be distinct by sequence but we need further
+guidance to make the global feature shared across all the event times in a sequence. In fact, due to
+the permutation invariance constraint (implemented in average-pooling), the global latent feature z
+cannot be very different at different event time: adding some addition context features ri will not
+change G as well as z much.
+For the latent variable z, additional guidance is provided, which is clearer with the objective of
+the variational inference. Recall that the objective of the variational inference in Equation (6) is
+provided as,
+arg max
+θ
+Ez∼pθ(z | CL) log pθ(τl | τl−k:l−1, z) − KL(pθ(z | CL) || pθ(z | Cl)).
+Here, regardless of the index of the target l, it always minimizes the KL divergence between
+pθ(z | CL) and pθ(z | Cl) where L is the length of a sequence.
+So, ideally, the latent variable
+z ∼ pθ(z | Cl) should capture the same information as z ∼ pθ(z | CL). It implies regardless of the
+index of the target l, the latent variable z asymptotically captures the global feature of the whole se-
+quence, which is equivalent to z ∼ pθ(z | CL). Hence, the resulting pθ(z | Cl) captures the global and
+task-specific patterns, which ideally is similar to pθ(z | CL). As a result, the global latent feature is
+guided to be distinct for different sequences but similar throughout different event time l ∈ [1, L−1]
+within the same sequence.
+It is also worth mentioning that its role is quite different from the target input τl−k+1:l which is
+another input for the decoder. Consider two events that are far apart from each other. Due to
+distinctive local patterns, their target inputs can be quite different from each other. On the other
+hand, the global features will not be that different as they are the average of all the context features
+at each event time, plus guided by the KL divergence. Hence, the global feature provides “overall”
+patterns of a task whereas a target input provides local patterns to the decoder.
+Back to our original goal: treating each sequence as a realization of a distinct stochastic process, we
+use the global latent feature that is distinct by each sequence to provide a task-specific information
+which is shared regardless of different event time step l. It is neither implicitly nor explicitly consid-
+ered in the supervised learning case. In supervised learning, each event time step at each sequence
+is treated equally from which patterns for only one stochastic process is learned.
+In the Table 5, we compare the THP+ baseline and Meta TPP on Sinusoidal, Uber and NYC Taxi
+datasets to demonstrate the effectiveness of the global latent feature. The decoder of the Meta TPP
+takes the global latent feature z (from the permutation invariance constraint) as an input, in addition
+to the target input feature r l that the decoder of the THP+ baseline takes as input. The result shows
+that the global latent feature generally helps to improve both RMSE and NLL performance.
+12
+
+Methods
+Sinusoidal
+Uber
+NYC Taxi
+RMSE
+NLL
+RMSE
+NLL
+RMSE
+NLL
+THP+
+1.72
+0.84
+90.25
+3.63
+10.31
+2.00
+Meta TPP
+1.48
+0.61
+63.35
+3.25
+10.04
+2.33
+Table 5: Comparison between THP+ baseline and Meta TPP.
+C
+COMPUTATION OF EVALUATION METRICS
+Unlike Equation (7) in the main paper where the ELBO is computed using samples from pθ(z | CL),
+in inference, we do not have access to z ∼ pθ(z | CL). But, as pθ(z | Cl) is trained to be similar
+to pθ(z | CL) through KL(pθ(z | CL) | pθ(z | Cl), we use samples z ∼ pθ(z | Cl). As specified in
+Appendix I, we use 256 samples to have good enough approximation.
+• NLL – We approximate a log-likelihood of the next event time τl+1 using Monte-Carlo
+approximation as,
+log pθ(τl+1 | τl−k+1:l, Cl) = log
+�
+pθ(τl+1 | τl−k+1:l, z)pθ(z | Cl)dz
+(11)
+≈ log 1
+M
+M
+�
+m=1
+pθ(τl+1 | τl−k+1:l, zm)
+(12)
+where M is the number of samples from pθ(z | Cl).
+• RMSE – We use a mixture of log-normal distributions to model pθ(τl+1 | τl−k+1:l, z). For-
+mally, for l ∈ [1, L − 1], τl+1 ∼ MixLogNorm(µl+1, σl+1, ωl+1) where µl+1 are the
+mixture means, σl+1 are the standard deviations, and ωl+1 are the mixture weights. The
+parameters are the outputs of the decoder given a latent sample z. Knowing this, we can
+analytically compute the expected event time for a latent sample z with K mixture compo-
+nents as,
+Eτl+1∼pθ(τl+1 | τl−k+1:l,z)[τl+1] =
+K
+�
+k=1
+ωl+1,k exp (µl+1,k + 1
+2σ2
+l+1,k).
+Note that since this expectation is over pθ(τl+1 | τl−k+1:l, z) where z is one sample from
+the posterior, we need to take another expectation over the posterior as follows,
+Eτl+1∼pθ(τl+1 | τl−k+1:l,Cl)[τl+1] = Ez∼pθ(z | Cl)Eτl+1∼pθ(τl+1 | τl−k+1:l,z)[τl+1]
+(13)
+= Ez∼pθ(z | Cl)
+K
+�
+k=1
+ωl+1,k exp (µl+1,k + 1
+2σ2
+l+1,k) (14)
+≈ 1
+M
+M
+�
+m=1
+K
+�
+k=1
+ωl+1,k exp (µl+1,k + 1
+2σ2
+l+1,k)
+(15)
+where M is the number of samples from pθ(z | Cl).
+• Accuracy – We obtain class predictions by taking argmax over the probability distribution
+of class labels as follows,
+arg max
+c∈[1,C]
+pθ(yl+1 | τl−k+1:l, yl−k+1:l, Cl)
+where C is the number of marks. The probability distribution of class labels is approxi-
+mated using MC samples as,
+pθ(yl+1 | τl−k+1:l, yl−k+1:l, Cl) =
+�
+pθ(yl+1 | rl, z)pθ(z | Cl)dz
+(16)
+≈ 1
+M
+M
+�
+m=1
+pθ(yl+1 | rl, zm)
+(17)
+where M is the number of samples from pθ(z | Cl).
+13
+
+D
+MONTE-CARLO APPROXIMATION VS. VARIATIONAL INFERENCE
+Amortized variational inference (VI) we described in Section 3 is not the only way to approximate
+the latent variable model. We can also use Monte-Carlo (MC) approximation, which is simpler and
+does not rely on a proxy like ELBO. It is formulated as,
+log
+�
+pθ(τl+1 | τl−k+1:l, z)pθ(z | Cl)dz ≈ log 1
+N
+N
+�
+n=1
+pθ(τl+1 | τl−k+1:l, zn)
+(18)
+Note that a sample zn in Equation (18) is drawn from p(zn|Cl), which is different from zn ∼
+p(zn|CL) in Equation (7). Foong et al. (2020); Dubois et al. (2020) report that MC approximation
+generally outperforms variational inference. In variational inference, as a model is trained with
+z ∼ pθ(z | CL), the samples z ∼ pθ(z | Cl) in test time are quite different from what the model has
+used as inputs: although KL(pθ(z | CL) | pθ(z | Cl)) forces pθ(z | CL) and pθ(z | Cl) close to each
+other, it is hard to make KL-divergence to be zero. Although MC approximation outperforms VI for
+the proposed Meta TPP, it is not the case for the Attentive TPP as shown in Table 6.
+Attention
+Latent
+Reddit
+Uber
+NYC Taxi
+VI
+MC
+RMSE
+NLL
+Acc
+RMSE
+NLL
+RMSE
+NLL
+
+
+
+0.13
+-0.39
+0.61
+63.35
+3.25
+10.04
+2.33
+
+
+
+0.11
+0.16
+0.61
+37.12
+3.22
+10.15
+2.00
+
+
+
+0.11
+0.03
+0.60
+21.87
+2.98
+8.92
+2.00
+
+
+
+0.13
+-0.05
+0.60
+22.38
+3.18
+9.10
+2.01
+Table 6: Comparison of the variants of Meta TPPs
+We conjecture it based on MC approximation better sharing role with the cross-attention path when
+compared to the VI approximation. More specifically, cross-attention forces a model to have similar
+features for repeating local history patterns. As it focuses on extracting features from the previous
+history, which is similar to what z ∼ pθ(z | Cl) contains, the latent and attentive path share a role
+in MC approximation. On the other hand, as the model is trained on the global latent feature z ∼
+pθ(z | CL) in VI, without focusing too much on the previous history due to the cross-attention, it
+may be able to utilize more diverse features. It would be an interesting future work to investigate
+the theoretical relationship between approximation methods and variants of Meta TPP.
+E
+NA¨IVE BASELINE
+Sometimes na¨ıve baselines can be stronger baselines than more sophisticated ones. To investigate
+if that is the case for TPPs, we implement a na¨ıve baseline that makes predictions based on median
+inter-event interval: ˆτ l + 1 = τ l + ∆τ median, l where ∆τ median, l is a median of the inter-
+event interval up to l-th event. We boostrap for 200 times on the test set to obtain the mean of RMSE
+metrics as with how we obtain the numbers for the other methods (NLLs are not available for the
+na¨ıve baseline). In Table 7, the performance of the na¨ıve baseline is surprisingly good for some
+cases. For instance, it is better than the intensity-free on Wiki and NYC Taxi datasets. It is, however,
+much worse than THP+ and the proposed Attentive TPP on all the datasets.
+Methods
+Stack Overflow
+Mooc
+Reddit
+Wiki
+Sinusoidal
+Uber
+NYC Taxi
+Na¨ıve baseline
+161.21
+0.79
+0.38
+0.21
+4.61
+107.91
+24.58
+Intensity-free
+3.64
+0.31
+0.18
+0.60
+1.29
+51.23
+46.59
+Neural flow
+–
+0.47
+0.32
+0.56
+1.13
+–
+–
+THP+
+1.68
+0.18
+0.26
+0.17
+1.72
+90.25
+10.31
+Attentive TPP
+1.15
+0.16
+0.11
+0.15
+1.45
+22.11
+8.92
+Table 7: Comparison of Na¨ıve baseline and other methods
+14
+
+F
+EFFECT OF MODEL SIZE
+Although we have demonstrated that the improvement in performance does not come from the size
+of a model through Table 4 on the Reddit dataset, we provide more evidence on the rest of the
+datasets as below.
+Methods
+# Params
+Sinusoidal
+Uber
+NYC Taxi
+RMSE
+NLL
+RMSE
+NLL
+RMSE
+NLL
+THP+
+50K
+1.72 (0.10)
+0.84 (0.02)
+90.25 (4.53)
+3.63 (0.03)
+10.31 (0.47)
+2.00 (0.01)
+100K
+1.84 (0.13)
+1.04 (0.02)
+82.69 (4.56)
+3.34 (0.03)
+10.16 (0.47)
+1.92 (0.01)
+Attentive TPP
+96K
+1.45 (0.11)
+0.66 (0.02)
+22.11 (1.94)
+2.89 (0.04)
+8.92 (0.42)
+2.00 (0.009)
+Table 8: Comparison of different model size on periodic datasets.
+Methods
+# Params
+Stack Overflow
+RMSE
+NLL
+Acc
+THP+
+52K
+1.68 (0.16)
+3.28 (0.02)
+0.46 (0.004)
+103K
+1.63 (0.06)
+2.82 (0.03)
+0.46 (0.004)
+Attentive TPP
+99K
+1.15 (0.02)
+2.64 (0.02)
+0.46 (0.004)
+Table 9: Comparison of different model size on the Stack Overflow dataset.
+Methods
+# Params
+Mooc
+RMSE
+NLL
+Acc
+THP+
+56K
+0.18 (0.005)
+0.13 (0.02)
+0.38 (0.004)
+113K
+0.22 (0.007)
+0.05 (0.03)
+0.39 (0.004)
+Attentive TPP
+108K
+0.16 (0.004)
+-0.72 (0.02)
+0.36 (0.003)
+Table 10: Comparison of different model size on the Mooc dataset.
+Methods
+# Params
+Wiki
+RMSE
+NLL
+Acc
+THP+
+577K
+0.17 (0.02)
+6.25 (0.39)
+0.23 (0.03)
+1153K
+0.16 (0.01)
+6.47 (0.40)
+0.21 (0.02)
+Attentive TPP
+1149K
+0.15 (0.01)
+6.25 (0.38)
+0.25 (0.03)
+Table 11: Comparison of different model size on the Wiki dataset.
+In the table above, we observe that in many cases, smaller models perform better than larger models :
+on Sinusoidal, Mooc, and Wiki. Although it is sometimes true that larger models perform better than
+their smaller counterparts, they are still significantly worse than our proposed Attentive TPP. Note
+that we conducted exactly the same grid search for hyperparameter tuning for the larger models. The
+results empirically demonstrate that the improvement does not necessarily come from the size of a
+model but from right inductive biases.
+Lastly, please note that all the experiment results we have reported in Table 1-4 and in rebuttal are
+on the test sets. We believe the RMSE, NLL, and Accuracy on test sets are good metrics to compare
+the generalization performance of different models. Given that our proposed method outperforms
+all the baselines, we think our experiments empirically demonstrate the robustness of our method in
+terms of generalization.
+G
+EXTENSION TO MARKED TPPS
+We extended the proposed method to the marked cases by adding class prediction branch following
+the intensity-free TPP (Shchur et al., 2020). Suppose a mark at l + 1-th event is denoted as yl+1.
+15
+
+Datasets
+# of Seq.
+# of Events
+Max Seq. Length
+# of Marks
+Stack Overflow
+6,633
+480,414
+736
+22
+Mooc
+7.047
+389,407
+200
+97
+Reddit
+10,000
+532,026
+100
+984
+Wiki
+1,000
+138,705
+250
+7,628
+Sinusoidal
+1,000
+107,454
+200
+1
+Uber
+791
+701,579
+2,977
+1
+NYC Taxi
+1,000
+1,141,379
+1,958
+1
+Table 12: Statistics of the datasets.
+For the proposed Meta TPP, we compute the log-likelihood of the mark as,
+log pθ(yl+1 | τl−k+1:l, yl−k+1:l, Cl) = log
+�
+pθ(yl+1 | τl−k+1:l, yl−k+1:l, z)pθ(z | Cl)dz.
+Note that Cl includes both event times and corresponding labels. For implementation, we added one
+fully connected layer that takes as input the same features for the decoder (that predicts the next
+event time), and outputs the logits for classification. A class prediction is made by taking argmax
+over the probability distribution which is approximated using Monte-Carlo samples as,
+pθ(yl+1 | τl−k+1:l, yl−k+1:l, Cl) ≈ 1
+M
+M
+�
+m=1
+pθ(yl+1 | rl, zm)
+Note that inputs τl−k+1:l and yl−k+1:l are encoded to rl. The class predictions to compute the
+accuracies reported throughout the experiments are made from this.
+H
+DESCRIPTION OF DATASETS
+We use 4 popular benchmark datasets: Stack Overflow, Mooc, Reddit, and Wiki, and 3 newly pro-
+cessed datasets: Sinusoidal wave, Uber and NYC Taxi. The statistics are provided in Table 12. To
+split the data into train, validation, and test, we follow the splits made in the previous works such as
+Shchur et al. (2020), and Yang et al. (2022). More specifically, we use 60%, 20%, and 20% split for
+train, validation, and test, respectively, for all the datasets following Shchur et al. (2020) except for
+Stack Overflow. For Stack Overflow, we follow the split made by Yang et al. (2022) and Du et al.
+(2016) where 4,777, 530, and 1,326 samples are assigned for train, validation, and test, respectively.
+For more detailed descriptions of the popular benchmark datasets, please refer to the original papers
+as described below or Section E.2 of Shchur et al. (2020).
+H.1
+BENCHMARK DATASETS
+Stack Overflow. It was first processed in (Du et al., 2016). We use the first folder of the dataset
+following (Shchur et al., 2020; Yang et al., 2022).
+Mooc. It consists of 7,047 sequences, each of which contains of action times an individual user of
+an online Mooc course. There are 98 categories.
+Reddit. It consists of 10,000 sequences from the most active users with marks being the sub-reddit
+categories of each sequence.
+Wiki. It consists of 1,000 sequences from the most edited Wikipedia pages (for a month period)
+with marks being users who made at least 5 changes.
+H.2
+PROPOSED DATASETS
+Sinusoidal wave. We generate the Sinusoidal wave using a sine function with a periodicity of 4π
+and the domain of [0, 32π]. We randomly choose the number of events per sequence in [20, 200] for
+1,000 sequences.
+16
+
+0
+20
+40
+60
+80
+100
+Unit
+0
+2
+4
+6
+Bin counts
+(a) An example in Sinusoidal wave dataset.
+0
+20
+40
+60
+80
+100
+120
+140
+Days
+0
+5
+10
+15
+20
+Bin counts
+(b) An example in Uber dataset.
+0
+50
+100
+150
+200
+250
+300
+350
+400
+Hours
+0
+2
+4
+6
+Bin counts
+(c) An example in NYC Taxi dataset.
+Figure 4: Visualization of examples in the datasets with strong periodicity.
+Uber.2 We generate the Uber dataset using the data from the shared link. Among the data from
+Januaray, 2015 to June, 2015, we create sequences using Dispatching-base-num and locationID as
+keys. We also give a constraint that the mininum and maximum events per sequence being 100 and
+3,000, respectively, and drop all the overlapping event times.
+NYC Taxi.3 We generate the NYC Taxi dataset from the NYC Taxi pickup raw data in 2013 shared
+in the link, which is different from the one proposed in Du et al. (2016), as we do not include
+any location information. We generate 6 different datasets by splitting the whole data in 2013 for
+every two months: Jan-Feb, Mar-Apr, May-Jun, Jul-Aug, Sep-Oct, and Nov-Dec. Throughout the
+experiment, we train models on the training set of Jan-Feb split, and evaluate on the test set of
+Jan-Feb for in-domin, and other for distribution drift.
+H.3
+PERIODICITY
+In Section 5.2, we provide experiment results that demonstrate the proposed Attentive TPP capture
+periodic patterns better than the baselines. To validate that the datasets used for Table 2 have strong
+periodic patterns, we provide visualization in Figure 4. Sinusoidal wave dataset has a periodicity for
+every 4π as shown in Figure 4a, Uber datset has weekly periodic pattern shown in Figure 4b, and
+NYC Taxi dataset has daily pattern as shown in Figure 4c.
+2https://www.kaggle.com/datasets/fivethirtyeight/uber-pickups-in-new-york-city/metadata
+3http://www.andresmh.com/nyctaxitrips/
+17
+
+I
+HYPERPARAMETERS
+We use the feature dimension of 96, 72, and 64 for the intensity-free (Shchur et al., 2020), neural
+flow (Biloˇs et al., 2021), and THP+, respectively, as the numbers of parameters fall in the range of
+50K and 60K with those dimensions.
+For the Meta TPP, we use 64 for the dimension of the latent variable z, and 32 samples to approx-
+imate the ELBO for variantional inference. As the variance of variational inference is generally
+low, 32 samples are enough to have stable results. In inference, we increase the sample size to 256
+to have more accurate approximation. For the Attentive TPP, we use 1-layer self-attention for the
+cross-attention path, and fix the local history window size to 20.
+For training, we use a batch size of 16 throughout all the models, and optimize with an Adam
+optimizer for a grid search for learning rate and weight decay described in Section 5.
+18
+

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+MIT–CTP 5475
+DESY–22–159
+Field-Theoretic Analysis of Hadronization Using Soft Drop Jet Mass
+Anna Ferdinand,1, 2, ∗ Kyle Lee,3, 4, † and Aditya Pathak1, 5, ‡
+1University of Manchester, School of Physics and Astronomy, Manchester, M13 9PL, United Kingdom
+2DAMTP, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, United Kingdom
+3Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
+4Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
+5Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany
+One of the greatest challenges in quantum chromodynamics is understanding the hadronization
+mechanism, which is also crucial for carrying out precision physics with jet substructure. In this
+Letter, we combine recent advancements in our understanding of the field theory-based nonperturba-
+tive structure of the soft drop jet mass with precise perturbative calculations of its multi-differential
+variants at next-to-next-to-leading logarithmic (NNLL) accuracy. This enables a systematic study
+of its hadronization power corrections in a completely model-independent way. We calibrate and
+test hadronization models and their interplay with parton showers by comparing our universality
+predictions with various event generators for quark and gluon initiated jets in both lepton-lepton
+and hadron-hadron collisions.
+We find that hadronization models perform better for quark jets
+relative to gluon jets. Our results provide the necessary toolkit for precision studies with the soft
+drop jet mass motivating future analyses using real world collider data. The nontrivial constraints
+derived in our framework are useful for improving the modeling of hadronization and its interface
+with parton showers in next generation event generators.
+The study of jets and their substructure has become a
+very active program at high energy particle colliders in
+the last decade [1, 2]. A key development has been the use
+of jet grooming techniques [3–10] that allow for theoreti-
+cal control by eliminating contamination from the wide-
+angle soft radiation from the underlying event and pile-
+up, and by reducing hadronization effects. In particular,
+the soft drop (SD) grooming [6–8] has received the most
+widespread attention, inspiring many theoretical calcula-
+tions both for jets in vacuum [11–38] and in medium [39–
+43], as well as several experimental analyses [44–53].
+Among various groomed observables, the SD jet mass
+is by far the most extensively studied within both theo-
+retical [6, 54–60] and experimental communities [61–71],
+and has been explored for a variety of phenomenologi-
+cal applications, such as quantifying medium modifica-
+tion [39, 61, 68], and precision top quark mass [72–74]
+and strong coupling constant [75, 76] measurements.
+Theoretically, the SD jet mass is the most precisely
+studied groomed jet observable, with predictions avail-
+able at next-to-next-to-next-to-leading logarithmic accu-
+racy (N3LL) matched to next-to-next-to-leading order
+(NNLO) predictions for dijets at e+e− collisions [77],
+and next-to-next-to-leading-logarithmic (NNLL) accu-
+racy [76] for jets at the LHC. At this level of precision,
+the hadronization power corrections become comparable
+in size to perturbative accuracy, and cannot be accounted
+for using hadronization models [78–80] that are tuned
+to lower precision parton showers. In the recent years,
+there has been significant progress in understanding these
+hadronization effects in the SD jet mass [81–83] using a
+field theory-based formalism [84–90], which allows for a
+model-independent description of nonperturbative (NP)
+power corrections for precision phenomenology. Further-
+more, this formalism imposes powerful constraints on jet-
+FIG. 1: An example of fit for nonperturbative param-
+eters in Pythia 8.306 simulation of groomed jet mass.
+The insets show distribution of low energy particles as
+heat maps around the soft drop stopping subjets in the
+transverse plane.
+flavor, kinematics and grooming parameter dependence
+of these NP corrections. Hence, by comparing these pre-
+dictions with event generators, we now have a unique
+opportunity to carry out a nontrivial characterization
+of these hadronization models and their interplay with
+parton showers, which is often difficult to interpret and
+test. In this Letter, using the state-of-the-art theoretical
+advancements in our understanding of the SD jet mass,
+we achieve a systematic and complete field theory-based
+study of hadronization effects and test our predictions
+with multiple event generators.
+Hadronization corrections to groomed jet mass.—
+Compared to the ungroomed jet mass, the SD jet mass
+exhibits a much larger region of applicability for pertur-
+arXiv:2301.03605v1  [hep-ph]  9 Jan 2023
+
+8C2
+bation theory. This region is referred to as the soft drop
+operator expansion (SDOE) region, which is defined be-
+low and shown in Fig. 1 between the vertical lines. Here,
+hadronization effects can be studied using factorization
+in a systematic expansion.
+Using soft collinear effective theory (SCET) [91–94],
+in Ref. [81] the leading hadronization corrections in the
+SDOE region were shown to depend on three O(ΛQCD)
+NP universal constants {Ω◦◦
+1κ, Υ�
+1,0κ, Υ�
+1,1κ}, which solely
+depend on the parton κ = q, g initiating the jet, and
+are completely independent of the jet kinematics, such
+as the jet pT (or EJ), rapidity ηJ, radius R, and the
+SD parameters [7], the energy cut zcut and the angular
+modulation parameter β, such that
+1
+σκ
+dσκ
+dm2
+J
+= 1
+ˆσκ
+dˆσκ
+dm2
+J
+− QΩ◦◦
+1κ
+d
+dm2
+J
+1
+ˆσκ
+dˆσ◦◦
+κ
+dm2
+J
+(1)
++ Υ�
+1,0κ + βΥ�
+1,1κ
+Q
+1
+ˆσκ
+dˆσ�
+κ
+dm2
+J
++ · · · ,
+Here dσκ and dˆσκ, respectively, refer to hadron and par-
+ton level groomed jet mass cross sections for flavor κ
+and Q characterizing the the hard scale of the jet. The
+weights dˆσ◦◦,�
+κ
+are perturbatively calculable.
+We note
+that, in contrast with analytical hadronization models
+employed in previous work [6, 60, 75, 89], Eq. (1) is a
+model-independent statement and includes hadron mass
+effects.
+In the SDOE region, the leading hadronization cor-
+rections are driven by a two-pronged dipole, which con-
+sists of an energetic collinear subjet at the core of the
+jet and a collinear-soft (c-soft) subjet that is responsible
+for stopping the grooming algorithm.
+The corrections
+represented by the ellipsis ‘. . .’ in Eq. (1) involve higher
+power corrections of ΛQCD and corrections from configu-
+rations that distort the two-pronged catchment area. The
+latter correction is a next-to-leading-logarithmic effect,
+and therefore Eq. (1) can also be seen as a factorization
+of NP effects at leading-logarithmic accuracy, where the
+strong ordering of angles ensures the two-pronged geom-
+etry. As the jet mass decreases, we enter the soft drop
+non-perturbative (SDNP) region, where the c-soft mode
+becomes nonperturbative and correspondingly the non-
+perturbative effects are of O(1). The transition between
+these two regions is clearly visible in Fig. 1, where the
+insets show the distribution of low-energy NP particles
+in the transverse plane of the jet [81].
+The statement of NP factorization in Eq. (1) presents
+us with a singular opportunity to probe hadronization in
+jets in a rich setting. As can be seen from Eq. (1), the
+consistency of the formalism requires that the three con-
+stants be sufficient to describe data measured from high
+energy colliders over a wide range of energies. The highly
+constraining structure given by Eq. (1) (constants being
+of O(ΛQCD), having a β proportional coefficient Υ�
+1,1κ,
+zcut-independence, etc.) makes this far from a trivial feat
+FIG. 2: NNLL results for perturbative weights in Eq. (1)
+of hadronization corrections (shown here for gluon jets).
+Bands denote perturbative uncertainty and vertical lines
+the extent of the fit region (see Eq. (7)).
+The factor
+of 1/Q is included to illustrate the size of hadronization
+corrections.
+and hence useful for calibrating hadronization models. In
+this work, we demonstrate how the universality structure
+strongly constrains the NP parameters, allowing them to
+be accurately determined by considering various combi-
+nations of soft drop and kinematic parameters. This, for
+example, improves the prospects for measuring the strong
+coupling constant αs at the LHC.
+Calculation of perturbative weights.— Characterizing
+the two-pronged configuration of the collinear and the
+c-soft subjet in the SDOE region requires auxiliary mea-
+surements of the groomed jet radius Rg and soft subjet
+energy fraction zg [7, 34, 35, 95], which after marginaliz-
+ing give [82]
+1
+ˆσκ
+dˆσ◦◦
+κ
+dm2
+J
+≡
+�
+drg rg
+1
+ˆσκ
+d2ˆσκ
+dm2
+Jdrg
+,
+(2)
+1
+ˆσκ
+dˆσ�
+κ
+dm2
+J
+≡
+� drgdzg δ
+�
+zg − zcutrβ
+g
+�
+rg
+1
+ˆσκ
+d3ˆσκ
+dm2
+Jdrgdzg
+.
+As the NP constants in Eq. (1) are independent of the
+jet kinematics and grooming parameters, all these depen-
+dencies are encapsulated by dˆσ◦◦,�
+κ
+. The appearance of
+rg = Rg/R in Eq. (2) is analogous to how jet radius R
+appears in hadronization corrections for the ungroomed
+jet mass in the tail and for the jet pT [89, 90]:
+m2
+J,no sd = ˆm2
+J,no sd+pT R Ω1κ ,
+pT = ˆpT + 1
+RΥ1κ , (3)
+where Ω1κ, Υ1κ ∼ ΛQCD are NP parameters and hatted
+variables are parton level values. In the case of the SD
+jet mass, the dynamically determined groomed jet radius
+Rg plays the role of R. The term in Eq. (1) with dˆσ◦◦
+κ is
+analogous to the ungroomed jet mass shift correction in
+
+3
+the tail, but is now described by a different constant Ω◦◦
+1κ
+as m2
+J = ˆm2
+J + pT RgΩ◦◦
+1κ.
+The term in the second line in Eq. (1) with dˆσ�
+κ is
+called the boundary correction. This effect is similar to
+the migration of events across pT -bins due to hadroniza-
+tion.
+Near the “boundary” of the c-soft subjet pass-
+ing/failing soft drop, i.e. when zg ≈ zcutrβ
+g , the partonic
+values ˆzg and ˆrg are modified due to hadronization as
+zg = ˆzg + 1
+rg
+Υ�
+1,0κ
+pT R ,
+rg = ˆrg − Υ�
+1,1κ
+pT R .
+(4)
+Here, Υ�
+1,0κ characterizes the shift in the pT of the c-
+soft subjet analogous to jet pT shift in Eq. (3), and Υ�
+1,1κ
+describes the change in the subjet location relative to the
+collinear subjet. The combination of the two gives rise
+to the linear structure Υ�
+1κ = Υ�
+1,0κ + βΥ�
+1,1κ as shown in
+Eq. (1), and constitutes a nontrivial prediction. Finally,
+it is useful to factor out the parton level groomed jet mass
+cross section from dσ◦◦,�:
+dˆσκ
+dm2
+J
+Cκ
+1 (m2
+J) ≡ dˆσ◦◦
+κ
+dm2
+J
+,
+dˆσκ
+dm2
+J
+Cκ
+2 (m2
+J) ≡ dˆσ�
+κ
+dm2
+J
+.
+(5)
+This definition is convenient as it will allow us to combine
+analytical calculation of the coefficients Cκ
+1,2(m2
+J) with
+parton shower jet mass cross section dˆσκ as discussed
+below.
+In Ref. [81], Cκ
+1,2(m2
+J) were computed in the coherent
+branching framework at LL accuracy. The first big step
+towards improving the accuracy of these coefficients was
+achieved in Ref. [82] by recasting them as moments of
+doubly differential cross section as in Eq. (2) and comput-
+ing them at NLL′ accuracy in the SDOE region. In this
+work, we employ a further improved calculation at NNLL
+accuracy described in the companion paper in Ref. [83],
+where the matching of the doubly differential cross sec-
+tion in the ungroomed region is included for correct treat-
+ment of the soft drop cusp location at NNLL. In Fig. 2,
+we show calculations of dˆσ◦◦,�
+κ
+for gluon jets at NNLL
+accuracy. With O(1 GeV) NP constants and kinematic
+prefactors as shown in Eq. (2), we see that the leading
+hadronization corrections can be as large as 10% for small
+jet masses.
+Calibrating hadronization models.— With state-of-the-
+art NNLL perturbative results for Cκ
+1,2(m2
+J), we are in po-
+sition to carry out a precise calibration of hadronization
+models.
+Furthermore, by incorporating NNLL pertur-
+bative uncertainty, we are able to significantly improve
+upon the analysis of Ref. [82] with LL predictions lack-
+ing uncertainty estimates.
+We simulate e+e− → gg,
+e+e− → q¯q, pp → Z + q jet and pp → Z + g jet pro-
+cesses using Pythia 8.306 [78], Vincia 2.3 [96] and
+Herwig 7.2.3 [80] parton showers with their default
+hadronization models. We reconstruct anti-kT [97] jets
+with R = 0.8 using Fastjet [98], and analyze them us-
+ing jet analysis software JETlib written by two of the
+2.4
+2.2
+2.0
+1.8
+1.6
+1.4
+1.2
+0.2
+0.4
+0.6
+0.8
+1.0
+C1( ), C2( )
+Partonic, e + e
+qq
+EJ = 500 GeV
+zcut = 0.1,
+= 1, R = 0.8
+=
+m2
+J
+Q2
+Partonic, e + e
+gg
+EJ = 500 GeV
+2.4
+2.2
+2.0
+1.8
+1.6
+1.4
+1.2
+log10
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+C1( ), C2( )
+Partonic, pp
+Z + q Jet
+450 < pT < 550, |
+J| < 2.5
+C1 NNLL
+C2 NNLL
+Pythia 8.306
+Herwig 7.2.3
+2.4
+2.2
+2.0
+1.8
+1.6
+1.4
+1.2
+1.0
+log10
+Partonic, pp
+Z + g Jet
+450 < pT < 550, |
+J| < 2.5
+FIG. 3: Weighted cross sections for hadronization cor-
+rections normalized to parton level jet mass spectrum as
+defined in Eq. (5) for zcut = 0.1 and β = 1.
+authors [99]. For e+e− collisions, we sample both jets
+in the dijet configuration, while only using the leading
+jet in pp collisions.
+As NP parameters are explictly
+predicted to be independent of the jet kinematics and
+grooming parameters, we carry out analysis using a wide
+range of kinematic and grooming parameter choices. In
+e+e− collisions, we analyze events at center of mass
+energies Q = 500, 750, 1000 GeV, while in pp, we use
+jets with pT
+∈ {[400, 600], [600, 800], [800, 1000]} GeV
+and soft drop parameters zcut ∈ {0.05, 0.1, 0.15, 0.2} and
+β ∈ {0, 0.5, 1, 1.5, 2}.
+We begin by explicitly defining the SDOE region where
+our analysis is carried out. We first define a dimensionless
+variable ξ ≡ m2
+J/Q2, where
+Q(pp) ≡ pT R ,
+Q(ee) ≡ 2EJ .
+(6)
+In terms of ξ, the SDOE region is then defined as ξ ∈
+�
+ξSDOE, ξ′
+0
+�
+, where
+ξSDOE ≡ ξ0
+�ρΛQCD
+Qξ0
+� 2+β
+1+β ,
+ξ′
+0 ≡
+ξ0
+(1 + ζ2)
+2+β
+2
+.
+(7)
+Here ξ0 is the location of the soft drop cusp [76, 83]:
+ξ(pp)
+0
+= zcut
+� R
+R0
+�β
+,
+ξ(ee)
+0
+= zcut
+�√
+2 tan R
+2
+sin R0
+2
+�β
+,
+(8)
+while ζ is defined by
+ζ(pp) ≡
+R
+2 cosh ηJ
+,
+ζ(ee) ≡ tan R
+2 ,
+(9)
+such that ξ′
+0 in Eq. (7) is the soft-wide angle transition
+point of the NNLL calculation. We set ΛQCD → 1 GeV,
+the typical scale of transition from parton showers to
+hadronization. The parameter ρ in Eq. (7) determines
+
+4
+Quark Jets Ω◦◦
+1q(GeV) Υ�
+1,0q(GeV) Υ�
+1,1q(GeV) χ2
+min/dof.
+e+e− →q¯q
+0.55+0.06
+−0.03
+−0.57+0.16
+−0.19
+1.06+0.31
+−0.35
+0.77+0.03
+−0.00
+pp→Z+q
+0.56+0.05
+−0.14
+−0.73+0.29
+−0.28
+0.89+0.27
+−0.25
+0.65+0.01
+−0.02
+Gluon Jets Ω◦◦
+1g(GeV) Υ�
+1,0g(GeV) Υ�
+1,1g(GeV) χ2
+min/dof.
+e+e− →gg
+1.92+0.16
+−0.32
+−0.48+0.23
+−0.22
+0.87+0.25
+−0.25
+3.13+0.05
+−0.20
+pp→Z+g
+0.93+0.01
+−0.12
+−0.24+0.11
+−0.01
+0.89+0.20
+−0.23
+1.34+0.05
+−0.10
+TABLE I: Fit results for NP constants in Pythia 8.306
+for quark and gluon jets in e+e− and pp collisions.
+the onset of the SDOE region, and we set ρ = 4.5. In
+principle, any choice satisfying ρ ≫ 1 is acceptable. We
+explore other choices of ρ in the Supplemental Material.
+In Fig. 3 we show a comparison of the NNLL compu-
+tation of Cκ
+1,2 with partonic Pythia and Herwig. The
+parton level results for from Vincia are found to be al-
+most identical to Pythia. We find a good agreement of
+the NNLL Cκ
+1 with MC for all four processes. The unusu-
+ally small errors for Cκ
+1 result from cancellation between
+correlated uncertainties in the two factors in Eq. (5). For
+pp, the agreement for the boundary term is poor for jet
+masses close to the cusp due to the initial-state radi-
+ation (ISR) contribution.
+However, as seen in Fig. 2,
+the NP corrections in the cusp-region are relatively sup-
+pressed, and NP corrections from ISR are also expected
+to be smaller as they involve subleading r2
+g moment of the
+boundary cross section [83]. Consequently, these effects
+do not significantly impact the analysis below.
+Finally, we perform a least-squares fit for the NP pa-
+rameters by defining our χ2 statistic as
+χ2 ≡
+�
+i
+�
+(⃗σMC
+κ,had)i − (⃗σκ,part+NP(Ω◦◦
+1κ, . . .)
+�
+i
+�2
+(∆⃗σ)2
+i
+.
+(10)
+Here, ⃗σX is a vector of cross section values for nbins = 10
+bins in the fit range and all permutations of pT (or
+EJ), zcut, and β values considered above.
+We denote
+the hadron level MC groomed jet mass cross section as
+⃗σMC
+κ,had, and define ⃗σκ,part+NP by including the NP con-
+stants Ω◦◦
+1κ, Υ�
+1,0κ, Υ�
+1,1κ and NNLL computation of Cκ
+1,2
+in Eq. (5) to the parton level MC spectrum dˆσMC
+κ
+fol-
+lowing Eq. (1). The uncertainty in the denominator is
+defined as
+(∆⃗σ)2
+i ≡
+�
+0.05(⃗σpart×C1)i
+�2 +
+�
+0.25(⃗σpart×C2)i
+�2 , (11)
+where, guided by the size of perturbative uncertainties
+in Fig. 2, we have assigned 5% and 25% uncertainty re-
+spectively to the weighted cross sections for shift and
+boundary corrections respectively.
+The NP constants
+Ω◦◦
+1κ, Υ�
+1,0κ, Υ�
+1,1κ are then varied to minimize this χ2
+statistic. An example of the fit for mass distribution is
+shown in Fig. 1.
+In Tab. I, we present the fit results for the NP con-
+stants with scale variations of Cκ
+1,2 for Pythia. As an-
+0.5
+1.0
+1.5
+2.0
+1 (GeV)
+1.6
+1.4
+1.2
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+1, 0 (GeV)
+Pythia 8.306
+e + e
+qq
+e + e
+gg
+pp
+Z + qJet
+pp
+Z + gJet
+C2 down
+C2 central
+C2 up
+0.5
+1.0
+1.5
+2.0
+1 (GeV)
+1.6
+1.4
+1.2
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+1, 0 (GeV)
+Vincia 2.3
+0.5
+1.0
+1.5
+2.0
+1 (GeV)
+Herwig 7.2.3
+FIG. 4: Testing jet flavor universality of soft drop NP pa-
+rameters in Pythia 8.306 (top), Vincia 2.3 (bottom,
+left) and Herwig 7.2.3 (bottom, right).
+ticipated, the parameters are ≲ 1 GeV. We also find
+similar parameter values for quark jets within the two
+quark processes within uncertainties.
+Even when NP
+parameters for quark jets are simultaneously fit for in
+e+e− and pp process, we find an excellent χ2 value of
+0.840/dof. This is expected, as soft drop isolates the jet
+from surrounding radiation. To further investigate this,
+we show correlations between Ω◦◦
+1κ and Υ�
+1,0κ for the four
+processes in Fig. 4 where each ellipse represents a 1σ
+deviation. To account for perturbative uncertainties, we
+repeat the fit by varying Cκ
+1,2 up and down within the un-
+certainty band shown in Fig. 3. We observe an excellent
+agreement within uncertainties between the NP param-
+eters for quark jets in pp and e+e− collisions in Pythia
+simulations, and a moderate agreement for Vincia and
+Herwig. In contrast, while Herwig exhibits similar levels
+of agreement for gluon jets and quark jets at both collid-
+ers, Pythia and Vincia show significant disagreement.
+This shows that contrary to the expectation for groomed
+jets, hadronization modeling of gluon jets in isolation in
+e+e− collisions in Pythia and Vincia differs significantly
+from jets in hadron colliders. Additionally, the differing
+results between Pythia and Vincia point to the inter-
+play of parton showers with hadronization models. In the
+Supplemental Material we show correlations in the other
+two combinations of NP parameters which show similar
+behavior as well as numerical fit results for Herwig and
+Vincia.
+Next, we test the grooming parameters independence
+
+5
+0.0
+0.5
+1.0
+1.5
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+1g (Gev)
+e + e
+gg
+Vincia 2.3
+Herwig 7.2.3
+Pythia 8.306
+0.0
+0.5
+1.0
+1.5
+2.0
+pp
+Z + gJet
+0.0
+0.5
+1.0
+1.5
+2.0
+1
+0
+1
+2
+3
+1q (Gev)
+e + e
+qq
+0.0
+0.5
+1.0
+1.5
+2.0
+pp
+Z + qJet
+FIG. 5: Test for linear β dependence of boundary correc-
+tions (Υ�
+1κ = Υ�
+1,0κ + βΥ�
+1,1κ) in gluon (top) and quark
+(bottom) jets for e+e− (left) and pp (right) collisions
+of these NP constants. We follow the same procedure
+as Ref. [81] and test this behavior by comparing the fit
+results for individual zcut and β values with the global
+fit. In Fig. 5, we demonstrate the linear β-dependence
+of the boundary correction by fitting for a single param-
+eter Υ�
+1κ(β) for each value of β. Because of degeneracy
+in the NP parameters, we fix Ω◦◦
+1κ to its global-fit value
+in this case. The error bars take into account perturba-
+tive uncertainty in Cκ
+1,2 by re-fitting with minimum and
+maximum variations. We find that all the three simula-
+tions perform well in each of the four cases. In Fig. 6,
+we repeat the same procedure to test zcut-independence
+of NP parameters.
+We find here that the three event
+generators pass the test for both quark and gluon jets in
+e+e− collisions, but exhibit a linear trend in zcut for both
+flavors in pp collisions. The larger χ2 values for gluon
+jets, as seen in Tab. I for Pythia (also true for Herwig
+and Vincia) suggest that modeling of hadronization in
+gluon jets is less consistent with our field theory predic-
+tions. Finally, our analysis of the e+e− → q¯q process
+using NNLL predictions of Cκ
+1,2 demonstrates significant
+improvement in the universality behavior of zcut and β,
+compared to Ref. [81] where LL predictions were used.1
+In conclusion, while our universality tests of the NP pa-
+rameters generally display expected behaviors in all the
+cases considered, they also reveal some tension with the
+1 Note that our numerical results for e+e− → q¯q also differ from
+those in Ref. [81] due to different prescription for error in Eq. (11)
+and newer versions of MC.
+0.05
+0.10
+0.15
+0.20
+zcut
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+1g (Gev)
+e + e
+gg
+Vincia 2.3
+Herwig 7.2.3
+Pythia 8.306
+0.05
+0.10
+0.15
+0.20
+zcut
+pp
+Z + gJet
+0.05
+0.10
+0.15
+0.20
+zcut
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1q (Gev)
+e + e
+qq
+0.05
+0.10
+0.15
+0.20
+zcut
+pp
+Z + qJet
+FIG. 6: Testing zcut-independence of Ω◦◦
+1κ for gluon (top)
+and quark jets (bottom) in e+e− (left) and pp collisions.
+hadronization models, pointing to interesting avenues for
+further improvement2 and motivate the use of real-world
+collider data for further analyses.
+Conclusions.— In this Letter, we have presented a sys-
+tematic framework for analyzing nonperturbative correc-
+tions in soft drop jet mass by bringing together earlier
+work on nonperturbative factorization and high preci-
+sion calculations of multi-differential soft drop cross sec-
+tions. Our analysis with hadronization models success-
+fully demonstrates that nonperturbative parameters ex-
+hibit the universal behaviors predicted by field theory.
+Our analysis is also directly applicable to precision phe-
+nomenology involving soft drop jet mass. For example,
+in Ref. [76] our results are used to assess the impact of
+the NP corrections on the sensitivity and ultimate pre-
+cision achievable on αs at the LHC using SD jet mass.
+Findings in Ref. [76] indicate that the hadronization ef-
+fects in the β = 1 case, for instance, are 3% (8%) for
+quark (gluon) jets when nonperturbative parameters in
+Eq. (1) are left unconstrained, which are of the same size
+as the NNLL perturbative uncertainty.
+We anticipate
+that with high precision calculations for the soft drop
+jet mass and the boundary correction (Cκ
+2 in Fig. 3),
+it will be possible to significantly constrain some or all
+of the NP constants, and hence improve the ultimate
+precision achievable on αs-determination at the LHC. In
+summary, our work thus provides crucial understanding
+2 For example, the analysis at LL in Reference [81] already revealed
+problems in the Herwig 8.2 hadronization model, which resulted
+in its improvement in version 8.3.
+
+6
+of hadronization corrections necessary for precision mea-
+surements with soft drop jet mass, a benchmark tool for
+improving hadronization modeling in MC event genera-
+tors, and motivation for analyses with real world collider
+data.
+Acknowledgements.— We would like to thank Mri-
+nal Dasgupta, Michael Seymour for helpful discussions.
+We are grateful to Simon Plätzer for many discus-
+sions and support with analysis with Herwig.
+We
+thank Holmfridur Hannesdottir, Johannes Michel and
+Iain Stewart for numerous discussions and feedback on
+the manuscript.
+We provide a numerical implementa-
+tion of the NNLL calculation in C++ building on core
+classes of SCETlib [100] which will be made available
+as a part of the scetlib::sd module [101]. We thank
+Johannes Michel for support with above-mentioned im-
+plementation in SCETlib.
+KL was supported by the
+LDRD program of LBNL and the U.S. DOE under con-
+tract number DE-SC0011090.
+AP acknowledges sup-
+port from DESY (Hamburg, Germany), a member of
+the Helmholtz Association HGF. AP was a member of
+the Lancaster-Manchester-Sheffield Consortium for Fun-
+damental Physics, which is supported by the UK Science
+and Technology Facilities Council (STFC) under grant
+number ST/T001038/1. AF also gratefully acknowledges
+support from the above-mentioned grant.
+∗ [email protected]
+† [email protected]
+‡ [email protected]
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+K.
+L.
+Michel,
+F.
+J.
+Tack-
+
+8
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+et
+al.,
+DESY-17-099
+(2018),
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+drop resummed observables in SCETlib,” (2022).
+
+1
+SUPPLEMENTAL MATERIAL
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+1g (Gev)
+e + e
+gg
+Vincia 2.3
+Herwig 7.2.3
+Pythia 8.306
+0.0
+0.5
+1.0
+1.5
+2.0
+pp
+Z + gJet
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+1q (Gev)
+e + e
+qq
+0.0
+0.5
+1.0
+1.5
+2.0
+pp
+Z + qJet
+FIG. 1: Testing β-independence of Ω◦◦
+1κ for quark and
+gluon jets while fixing Υ�
+1,0κ and Υ�
+1,1κ paramaters to
+their global-fit values.
+Quark Jets Ω◦◦
+1q(GeV) Υ�
+1,0q(GeV) Υ�
+1,1q(GeV) χ2
+min/dof.
+e+e− →q¯q
+0.62+0.39
+−0.10
+−0.69+0.05
+−0.16
+1.21+0.39
+−0.48
+0.94+0.06
+−0.01
+pp→Z+q
+0.93+0.16
+−0.14
+−1.05+0.26
+−0.26
+0.87+0.32
+−0.30
+0.82+0.02
+−0.02
+Gluon Jets Ω◦◦
+1g(GeV) Υ�
+1,0g(GeV) Υ�
+1,1g(GeV) χ2
+min/dof.
+e+e− →gg
+2.17+0.18
+−0.39
+−0.81+0.36
+−0.34
+1.16+0.34
+−0.32
+3.56+0.07
+−0.20
+pp→Z+g
+1.18+0.03
+−0.20
+−0.47+0.18
+−0.03
+1.06+0.24
+−0.28
+1.75+0.08
+−0.10
+TABLE I: Fit results for NP constants in Vincia 2.3.
+Quark Jets Ω◦◦
+1q(GeV) Υ�
+1,0q(GeV) Υ�
+1,1q(GeV) χ2
+min/dof.
+e+e− →q¯q
+0.34+0.25
+−0.03
+−0.64+0.09
+−0.19
+1.40+0.42
+−0.51
+0.63+0.10
+−0.02
+pp→Z+q
+0.66+0.17
+−0.08
+−0.98+0.24
+−0.29
+1.21+0.41
+−0.42
+0.27+0.03
+−0.01
+Gluon Jets Ω◦◦
+1g(GeV) Υ�
+1,0g(GeV) Υ�
+1,1g(GeV) χ2
+min/dof.
+e+e− →gg
+1.64+0.07
+−0.15
+−0.93+0.32
+−0.32
+1.09+0.34
+−0.33
+1.70+0.02
+−0.02
+pp→Z+g
+1.32+0.07
+−0.20
+−0.76+0.24
+−0.14
+0.94+0.23
+−0.26
+0.71+0.07
+−0.03
+TABLE II:
+Fit results for NP constants in Herwig
+7.2.3.
+In this Supplemental material, we present additional
+results for the calibration exercise of the MC hadroniza-
+tion model. Tables I and II present the results of fitting
+to Vincia 2.3 and Herwig 7.2.3, respectively. Similar
+to fits to Pythia 8.306 in Tab. I discussed in the main
+text, the χ2 values in Tab. I and Tab. II show that the
+fits for quark jets are much better constrained than gluon
+jets for both Vincia and Herwig, with those of Herwig
+0.05
+0.10
+0.15
+0.20
+zcut
+1.75
+1.50
+1.25
+1.00
+0.75
+0.50
+0.25
+0.00
+0.25
+1, 0g (Gev)
+e + e
+gg
+0.05
+0.10
+0.15
+0.20
+zcut
+pp
+Z + gJet
+0.05
+0.10
+0.15
+0.20
+zcut
+1.8
+1.6
+1.4
+1.2
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+1, 0q (Gev)
+e + e
+qq
+Vincia 2.3
+Herwig 7.2.3
+Pythia 8.306
+0.05
+0.10
+0.15
+0.20
+zcut
+pp
+Z + qJet
+FIG. 2: Testing zcut-independence of Υ�
+1,0κ for quark and
+gluon jets while fixing Υ�
+1,1κ and Ω◦◦
+1κ paramaters to their
+global-fit values.
+0.05
+0.10
+0.15
+0.20
+zcut
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+1, 1g (Gev)
+e + e
+gg
+0.05
+0.10
+0.15
+0.20
+zcut
+pp
+Z + gJet
+0.05
+0.10
+0.15
+0.20
+zcut
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+1, 1q (Gev)
+e + e
+qq
+Vincia 2.3
+Herwig 7.2.3
+Pythia 8.306
+0.05
+0.10
+0.15
+0.20
+zcut
+pp
+Z + qJet
+FIG. 3: Testing zcut-independence of Υ�
+1,1κ for quark and
+gluon jets while fixing Υ�
+1,0κ and Ω◦◦
+1κ paramaters to their
+global-fit values.
+being more consistent with our predictions.
+In Figs. 1, 2 and 3 we show tests for the independence
+of the grooming parameters Ω◦◦
+1κ, Υ�
+1,0κ, and Υ�
+1,1κ. The
+horizontal lines in these figures represent the global-fit
+values, while the markers with error bars represent fits for
+individual zcut or β parameters. We observe that for each
+process the results of fits to individual zcut or β values are
+arXiv:2301.03605v1  [hep-ph]  9 Jan 2023
+
+2
+0.5
+1.0
+1.5
+2.0
+1 (GeV)
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+1, 1 (GeV)
+Pythia 8.306
+e + e
+qq
+e + e
+gg
+pp
+Z + qJet
+pp
+Z + gJet
+C2 down
+C2 central
+C2 up
+1.4
+1.2
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+1, 0 (GeV)
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+1, 1 (GeV)
+Pythia 8.306
+0.5
+1.0
+1.5
+2.0
+1 (GeV)
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+1, 1 (GeV)
+Vincia 2.3
+1.4
+1.2
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+1, 0 (GeV)
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+1, 1 (GeV)
+Vincia 2.3
+0.5
+1.0
+1.5
+2.0
+1 (GeV)
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+1, 1 (GeV)
+Herwig 7.2.3
+1.4
+1.2
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+1, 0 (GeV)
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+1, 1 (GeV)
+Herwig 7.2.3
+FIG. 4: 1σ contours showing correlations between Υ�
+1,1κ and Ω◦◦
+1κ (left column) and between Υ�
+1,1κ and Υ�
+1,0κ (right
+column) for Pythia 8.306 (top row), Vincia 2.3 (middle row) and Herwig 7.2.3 (bottom row).
+consistent with the global-fit value within uncertainty,
+with the exception of β = 0 result of Ω◦◦
+1g for gluon jets
+in Fig. 1 where we notice a significant deviation.
+In order to better visualize the level of agreement for
+quark and gluon jets in e+e− and pp collisions for the
+three MC event generators considered we show in Fig. 4
+the 1σ contours for correlations between Υ�
+1,1κ and Ω◦◦
+1κ,
+and between Υ�
+1,0κ and Υ�
+1,1κ. The dominant uncertainty
+results from variation in Cκ
+2 . Here we see for all three
+hadronization models a better agreement of correlations
+between Ω◦◦
+1κ and Υ�
+1,0κ for quark jets in the two collid-
+ers than Υ�
+1,0κ and Υ�
+1,1κ. As noted earlier in the main
+text, Pythia and Vincia results for gluon jets for the
+two collider settings significantly disagree, whereas the
+corresponding results for Herwig display similar trend as
+quark jets.
+
+3
+3.0
+3.5
+4.0
+4.5
+5.0
+5.5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1q (Gev)
+e + e
+qq
+Vincia 2.3
+Herwig 7.2.3
+Pythia 8.306
+3.0
+3.5
+4.0
+4.5
+5.0
+5.5
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+2.25
+2.50
+2/d. o. f
+e + e
+qq
+Vincia 2.3
+Herwig 7.2.3
+Pythia 8.306
+3.0
+3.5
+4.0
+4.5
+5.0
+5.5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1q (Gev)
+pp
+Z + qJet
+Vincia 2.3
+Herwig 7.2.3
+Pythia 8.306
+3.0
+3.5
+4.0
+4.5
+5.0
+5.5
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+2/d. o. f
+pp
+Z + qJet
+Vincia 2.3
+Herwig 7.2.3
+Pythia 8.306
+3.0
+3.5
+4.0
+4.5
+5.0
+5.5
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+1g (Gev)
+pp
+Z + gJet
+Vincia 2.3
+Herwig 7.2.3
+Pythia 8.306
+3.0
+3.5
+4.0
+4.5
+5.0
+5.5
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+2/d. o. f
+pp
+Z + gJet
+Vincia 2.3
+Herwig 7.2.3
+Pythia 8.306
+FIG. 5: Fit range dependence of the fit values of NP parameters and the reduced χ2 parameterized in terms of ρ
+defined in Eq. (1).
+Finally, we now investigate the dependence of the fit
+values of the NP parameters and the resulting χ2 on the
+fit range, parameterized in terms of a parameter ρ defined
+via the equation
+ξSDOE ≡ ξ0
+�ρΛQCD
+Qξ0
+� 2+β
+1+β .
+(1)
+Thus, ρ determines the extent of the SDOE region. In
+Fig. 5 we illustrate the fit value obtained for Ω◦◦
+1κ (other
+parameters not shown for simplicity) and the correspond-
+ing reduced χ2 value for e+e− → q¯q and pp → Z + q, g
+jet processes. As the value of ρ increases, Eq. (1) implies
+that we move further into the perturbative region, which
+leads to a smaller errors on the fit value of Ω◦◦
+1κ and the
+
+4
+other two NP parameters due to the reduction in pertur-
+bative uncertainty on Cκ
+1,2 moments. At the same time,
+increasing ρ also reduces the available fit range which
+can impact the quality of the fit.
+As a result, we see
+in Fig. 5 that the reduced χ2 value initially improves as
+ρ is increased but eventually saturates as the fit range
+becomes too small. In order to find an optimal balance
+between perturbative uncertainty and fit range, we used
+the e+e− → q¯q process as a benchmark due to the ab-
+sence of ISR effects. We chose ρ = 4.5 as in all the results
+presented above, it is at this point the errors and χ2 val-
+ues stabilize while still allowing for a reasonable fit range.
+We also see this trend reflected in the values of Ω◦◦
+1κ for
+the e+e− → q¯q case. In the pp cases, however, we ob-
+serve a linear dependence of Ω◦◦
+1κ on the fit range, which
+is within the uncertainty for quark jets but not for gluon
+jets.
+

SNAzT4oBgHgl3EQf0f5U/content/tmp_files/2301.01784v1.pdf.txt

@@ -0,0 +1,1934 @@
+MNRAS 000, 1–16 (2023)
+Preprint 6 January 2023
+Compiled using MNRAS LATEX style file v3.0
+Local positive feedback in the overall negative: the impact of quasar winds
+on star formation in the FIRE cosmological simulations
+Jonathan Mercedes-Feliz,1⋆ Daniel Anglés-Alcázar,1,2 Christopher C. Hayward,2 Rachel K. Cochrane,2,3
+Bryan A. Terrazas,4 Sarah Wellons,5,8 Alexander J. Richings,6,7 Claude-André Faucher-Giguère,8
+Jorge Moreno,9 Kung Yi Su,2,10,14 Philip F. Hopkins,11 Eliot Quataert,12 and Dušan Kereš13
+1Department of Physics, University of Connecticut, 196 Auditorium Road, U-3046, Storrs, CT 06269-3046, USA
+2Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, New York NY 10010, USA
+3Harvard-Smithsonian Center for Astrophysics, 60 Garden St, Cambridge, MA 02138, USA
+4Columbia Astrophysics Laboratory, Columbia University, 550 West 120th Street, New York, NY 10027, USA
+5Department of Astronomy, Van Vleck Observatory, Wesleyan University, 96 Foss Hill Drive, Middletown, CT 06459, USA
+6E. A. Milne Centre for Astrophysics, Department of Physics and Mathematics, University of Hull, Cottingham Road, Hull, HU6 7RX, UK
+7DAIM, University of Hull, Cottingham Road, Hull, HU6 7RX, UK
+8CIERA and Department of Physics and Astronomy, Northwestern University, 1800 Sherman Ave., Evanston, IL 60201, USA
+9Department of Physics and Astronomy, Pomona College, 333 N. College Way, Claremont, CA 91711, USA
+10Department of Astronomy, Columbia University, 550 West 120th Street, New York, NY 10027, USA
+11TAPIR, Mailcode 350-17, California Institute of Technology, Pasadena, CA 91125, USA
+12Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA
+13Department of Physics, Center for Astrophysics and Space Sciences,University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA
+14Black Hole Initiative, Harvard University, 20 Garden St., Cambridge, MA 02138, USA
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+Negative feedback from accreting supermassive black holes is regarded as a key ingredient in suppressing star formation and
+quenching massive galaxies. However, several models and observations suggest that black hole feedback may have a positive
+effect, triggering star formation by compressing interstellar medium gas to higher densities. We investigate the dual role of black
+hole feedback using cosmological hydrodynamic simulations from the Feedback In Realistic Environments (FIRE) project,
+including a novel implementation of hyper-refined accretion-disc winds. Focusing on a massive, star-forming galaxy at z ∼ 2
+(Mhalo ∼ 1012.5 M⊙), we show that strong quasar winds with kinetic power ∼1046 erg/s acting for >20 Myr drive the formation of
+a central gas cavity and can dramatically reduce the star formation rate surface density across the galaxy disc. The suppression of
+star formation is primarily driven by reducing the amount of gas that can become star-forming, compared to directly evacuating
+the pre-existing star-forming gas reservoir (preventive feedback dominates over ejective feedback). Despite the global negative
+impact of quasar winds, we identify several plausible signatures of local positive feedback, including: (1) spatial anti-correlation
+of wind-dominated regions and star-forming clumps, (2) higher local star formation efficiency in compressed gas near the edge
+of the cavity, and (3) increased local contribution of outflowing material to star formation. Stars forming under the presence of
+quasar winds tend to do so at larger radial distances. Our results suggest that positive and negative AGN feedback can coexist
+in galaxies, but local positive triggering of star formation plays a minor role in global galaxy growth.
+Key words: galaxies: evolution – galaxies: star formation – quasars: general – quasars: supermassive black holes
+1 INTRODUCTION
+A copious amount of evidence show that most galaxies host a mas-
+sive black hole (BH) at their centre, with the mass of the BH strongly
+correlating with global galaxy properties (Magorrian et al. 1998;
+Bennert et al. 2011; Kormendy & Ho 2013; McConnell & Ma 2013;
+Reines & Volonteri 2015; Graham 2016; Shankar et al. 2020). Ac-
+tively accreting BHs in active galactic nuclei (AGN) can impact
+the host galaxy through a variety of feedback mechanisms, includ-
+⋆ E-mail: [email protected]
+ing fast accretion-driven winds (Faucher-Giguère & Quataert 2012;
+Zubovas & Nayakshin 2012; Tombesi et al. 2013; Nardini et al.
+2015), galaxy-scale outflows (Feruglio et al. 2010; Sturm et al. 2011;
+Greene et al. 2012; Cicone et al. 2014; Zakamska & Greene 2014;
+Wylezalek et al. 2020; Ramos Almeida et al. 2022), and large scale
+jets (Fabian 2012). Observational constraints on the efficiency of
+AGN feedback suggest that massive BHs may play a key role in
+galaxy evolution by injecting energy and momentum into the inter-
+stellar medium (ISM) and circumgalactic medium (CGM) of galax-
+ies (Hopkins & Elvis 2010; Alexander & Hickox 2012; Fabian 2012;
+Alatalo et al. 2015; Wylezalek & Zakamska 2016; Fiore et al. 2017;
+Harrison 2017; Harrison et al. 2018). Most galaxy formation models
+© 2023 The Authors
+arXiv:2301.01784v1  [astro-ph.GA]  4 Jan 2023
+
+2
+J. Mercedes-Feliz et al.
+indeed require some form of negative AGN feedback to eject exist-
+ing ISM gas from massive galaxies and/or prevent CGM gas from
+cooling and accreting onto the galaxy, suppressing star formation
+and quenching massive galaxies, regulating their sizes and central
+densities, and reproducing the observed bimodality in galaxy colours
+(Di Matteo et al. 2005; Baldry et al. 2006; Bower et al. 2006; Croton
+et al. 2006; Dubois et al. 2012; Silk & Mamon 2012; Somerville &
+Davé 2015; Choi et al. 2018; Davé et al. 2019; Wellons et al. 2022).
+However, recent observations suggest that AGN feedback could
+also have positive effects, by triggering star formation, as opposed
+to suppressing it, in galaxies. Direct observational evidence of AGN
+outflows triggering star formation is slim, but examples exist where
+star formation seems to occur within the outflow itself. Maiolino
+et al. (2017) studied a merging system that hosts an obscured AGN
+with a prominent outflow and found that multiple optical and near-
+infrared (IR) diagnostics of the outflowing gas are consistent with
+star formation within the outflow, where the inferred star formation
+rate (SFR) can exceed 15 M⊙ yr−1 and account for ∼25% of the to-
+tal SFR in the system. Analyzing over 2,500 galaxies in MaNGA
+DR2, Gallagher et al. (2019) identified a subsample of 37 galaxies
+with outflows, of which ∼30% show signs of star formation within
+the outflowing gas, ranging from 0.1–1 M⊙ yr−1 and contributing 5–
+30% of the total SFR in the galaxy. Positive and negative AGN feed-
+back do not necessarily act against one another, and some obser-
+vations suggest that their effects could simultaneously be present
+within the same galaxy. Cresci et al. (2015b) used SINFONI near-
+IR integral field spectroscopy of an obscured quasar at z ∼ 1.6 to
+show that a prominent outflow traced by [OIII] lines coincides with
+the location of an empty central cavity surrounded by star-forming
+regions, suggesting that the outflow is removing gas from the cav-
+ity (negative feedback) while triggering star formation at the edge
+of the cavity (positive feedback). Additional plausible implications
+of positive AGN feedback include the alignment of non-thermal ra-
+dio emission and rest-frame UV continuum emission in radio galax-
+ies, suggesting jet-induced star formation in the host galaxy (Bick-
+nell et al. 2000; Zirm et al. 2005; Drouart et al. 2016), higher SFR
+in green-valley galaxies with X-ray detected AGN and far infrared
+emission (Mahoro et al. 2017), and large scale expanding bubbles
+powered by jets potentially triggering star formation in other galax-
+ies (Gilli et al. 2019).
+Several theoretical models have explored the conditions under
+which AGN feedback can trigger star formation by compressing in-
+terstellar medium gas to higher densities, using analytic calculations
+(Begelman & Cioffi 1989; Rees 1989; Natarajan et al. 1998; King
+2005; Silk 2005; Ishibashi & Fabian 2012; Silk 2013; Zubovas et al.
+2013; Nayakshin 2014) and hydrodynamic simulations of idealized
+systems (Gaibler et al. 2012; Zubovas & Nayakshin 2012; Bieri et al.
+2015, 2016; Zubovas & Bourne 2017). Some of these models pro-
+pose that AGN feedback triggering of star formation plays a key
+role driving simultaneous AGN and star formation in galaxies (King
+2005), the observed correlation between star formation and AGN lu-
+minosities (Zubovas et al. 2013), the similarity in the comoving BH
+accretion rate density and the cosmic star formation history (Silk
+2013), the BH–galaxy scaling relations (Nayakshin 2014), the ex-
+treme SFRs of high redshift starbursts (Silk 2005; Gaibler et al.
+2012; Bieri et al. 2015, 2016), the size and structural evolution of
+massive galaxies (Ishibashi & Fabian 2012, 2014), and the forma-
+tion of dark matter-deficient dwarf galaxies from swept up gas in the
+intergalactic medium by quasar outflows (Natarajan et al. 1998, but
+see Moreno et al. 2022). Given the plausible strong implications for
+galaxy evolution predicted by these idealized models, it is crucial to
+investigate the impact of positive AGN feedback in more realistic
+simulations of galaxy formation in a full cosmological context.
+Large-volume cosmological hydrodynamic simulations, however,
+generally do not predict positive AGN feedback scenarios, in part by
+construction because their subgrid AGN feedback models are im-
+plemented to help suppress star formation and regulate the growth
+of massive galaxies when stellar feedback is not sufficiently strong
+(see the review by Somerville & Davé 2015, and references therein).
+In this context, the connection between AGN and star formation ac-
+tivity in galaxies is more naturally explained by a common gas sup-
+ply for star formation and BH growth (Anglés-Alcázar et al. 2015;
+Volonteri et al. 2015; Anglés-Alcázar et al. 2017a; Ricarte et al.
+2019; Thomas et al. 2019), and the extreme SFRs of high redshift
+galaxies are primarily driven by the systematically higher cosmolog-
+ical gas accretion rate onto halos and/or higher incidence of galaxy
+mergers (Davé et al. 2010; Hayward et al. 2013; Narayanan et al.
+2015), without requiring AGN feedback-driven triggering of star for-
+mation. However, cosmological simulations generally lack the reso-
+lution to model in detail the interaction of AGN-driven winds or jets
+with the multi-phase interstellar medium (ISM).
+In this paper, we investigate the plausible dual role of AGN
+feedback in galaxies using high-resolution cosmological zoom-in
+simulations from the Feedback In Realistic Environments (FIRE1)
+project (Hopkins et al. 2014, 2018), including local stellar feed-
+back by supernovae, stellar winds, and radiation in a multi-phase
+ISM, and a novel implementation of hyper-refined AGN-driven
+winds that simultaneously captures their propagation and impact
+from the inner nuclear region (≲10 pc) to circumgalactic medium
+(CGM) scales (Anglés-Alcázar et al., in prep.). Focusing on a mas-
+sive, star-forming galaxy near the peak of cosmic activity (z ∼ 2;
+Madau & Dickinson 2014), we investigate its subsequent evolution
+over a ∼35 Myr period in simulations with different AGN feedback
+strengths compared to that of an identical control simulation with-
+out AGN feedback. This provides an ideal framework to evaluate
+any positive versus negative feedback effects on the host galaxy.
+The outline of this paper is as follows: §2 provides a brief sum-
+mary of our methodology and §3 presents an overview of our sim-
+ulations. In §4 we investigate the global negative impact of AGN
+feedback on our simulated galaxy while §5 investigates plausible
+signatures of positive AGN feedback. In §6 we analyse the impact
+of AGN winds in simulations that vary the kinetic feedback effi-
+ciency. We discuss our results in §7 and present our summary and
+conclusions in §8.
+2 METHODS
+Anglés-Alcázar et al. (in prep.) fully describes the simulations and
+methodology that we implement, which we briefly summarize be-
+low.
+2.1 FIRE-2 galaxy formation model
+The simulations are part of the FIRE-2 project, an updated imple-
+mentation of the original FIRE simulations. The GIZMO2 solver is
+used in its meshless finite mass (MFM) mode, which implements a
+Lagrangian Godunov formulation with many of the benefits of par-
+ticle and grid-based methods (Hopkins 2015). We include cooling
+1 http://fire.northwestern.edu
+2 http://www.tapir.caltech.edu/~phopkins/Site/GIZMO.html
+MNRAS 000, 1–16 (2023)
+
+Local positive AGN feedback in the overall negative
+3
+1 kpc
+log10(
+SFR) [M yr
+1 kpc
+2]
+0
+2
+4
+log10(
+wind) [M kpc
+2]
+5
+6
+7
+Figure 1. Projected star formation rate surface density (ΣSFR) for the central 1 kpc region of a massive, star-forming galaxy (Mstar ∼ 1011 M⊙, SFR ∼
+300 M⊙ yr−1) at z ∼ 2.28 for the no-wind (top rows) and AGN-wind (bottom rows) simulations. In both cases we show edge-on and face-on views along
+with time evolution (from left to right) for ∼35 Myr since the start (∆t = 0) of the quasar feedback phase in the AGN-wind simulation. The projected mass
+density distribution of AGN winds is overlaid in the bottom rows, as indicated by the colour scale. In the absence of AGN-driven winds, the star-forming
+disc becomes denser and more compact as time progresses. In contrast, AGN winds evacuate star-forming gas from the central region, with the formation of a
+growing central cavity and global suppression of star formation by the end of the simulation.
+Name
+ηk
+ϵk
+˙Mw [M⊙ yr−1]
+˙Ew [erg s−1]
+Notes
+noAGN
+-
+-
+-
+-
+no-wind
+m0.1e0.5
+0.1
+0.005
+2.22
+6.29 × 1044
+m1e5
+1
+0.05
+22.2
+6.29 × 1045
+m2e10
+2
+0.1
+44.4
+1.26 × 1046
+AGN-wind
+m4e20
+4
+0.2
+88.8
+2.52 × 1046
+m10e50
+10
+0.5
+222
+6.29 × 1046
+Table 1. Simulation parameters: (1) Name: simulation designation. (2) ηk ≡
+˙Mw/ ˙MBH: mass loading factor. (3) ϵk ≡ ˙Ew/Lbol: kinetic feedback efficiency.
+(4) ˙Mw: mass outflow rate in winds. (5) ˙Ew: kinetic energy injection rate. (6)
+Notes.
+and heating from T = 10 − 1010 K; star formation in locally self-
+gravitating, dense (nH > 1000 cm−3), molecular, and Jeans-unstable
+gas; and stellar feedback from OB & AGB mass-loss, Type Ia & II
+Supernovae (SNe), and multi-wavelength photo-heating and radia-
+tion pressure (Hopkins et al. 2018). Each star particle represents a
+population of stars with known mass, age, and metallicity, with all
+stellar feedback quantities and their time dependence taken from the
+starburst99 population synthesis model (Leitherer et al. 1999).
+2.2 Initial conditions
+Our initial conditions are derived from snapshots of pre-existing
+FIRE-2 simulations, and adopted to include AGN-driven winds in
+our new simulations. We focus on the massive FIRE-2 halo A4 from
+Anglés-Alcázar et al. (2017c), with Mhalo ∼ 1012.5 M⊙ at z = 2
+and evolved down to z = 1 including on-the-fly BH growth driven
+by gravitational torques (Hopkins & Quataert 2011; Anglés-Alcázar
+et al. 2013, 2015, 2017a) but not including BH feedback. The new
+simulations with AGN winds adopt the same baryonic mass resolu-
+tion mb = 3.3 × 104 M⊙ and force softenings ϵmin
+gas = 0.7 pc, ϵ⋆ = 7 pc
+and ϵDM = 57 pc for the gas (minimum adaptive force softening),
+stellar, and dark matter components. We assume a ΛCDM cosmol-
+ogy with parameters H0 = 69.7 km s−1 Mpc−1, ΩM = 1 − ΩΛ =
+0.2821, Ωb = 0.0461, σ8 = 0.817, and ns = 0.9646 (Hinshaw et al.
+2013).
+We choose to inject AGN winds in the new simulations at the
+z = 2.28 snapshot, which will be referenced hereafter as ∆t = 0 Myr.
+At this time, the galaxy is undergoing a strong starburst phase which
+will lead to the formation of an overcompact and overdense stellar
+system because stellar feedback is no longer able to regulate star for-
+mation (Wellons et al. 2020; Parsotan et al. 2021, Anglés-Alcázar et
+al. in prep., Cochrane et al. in prep.). Cosmological hyper-refinement
+simulations of this galaxy have shown explicitly that strong gravita-
+tional torques from stellar non-axisymmetries can drive large gas
+inflow rates down to sub-pc scales under these conditions (Anglés-
+MNRAS 000, 1–16 (2023)
+
+4
+J. Mercedes-Feliz et al.
+1
+2
+3
+log10(vr) [km s
+1]
+0.2 Myr
+Outflow
+5.0 Myr
+10.0 Myr
+15.0 Myr
+20.0 Myr
+25.0 Myr
+35.0 Myr
+2
+1
+0
+log10(R) [kpc]
+0
+1
+2
+3
+42%
+Inflow
+62%
+40%
+55%
+60%
+55%
+52%
+1
+2
+3
+log10(vr) [km s
+1]
+Outflow
+2
+1
+0
+log10(R) [kpc]
+0
+1
+2
+3
+43%
+Inflow
+55%
+51%
+75%
+89%
+14%
+74%
+Figure 2. Two-dimensional, SFR-weighted distribution of radial velocity and radial distance for gas in the central ∼3 kpc for the no-wind (top rows) and AGN-
+wind (bottom rows) simulations. Time evolution is shown from left to right for ∼35 Myr (the colour scale is logarithmic and uniform throughout). In both cases
+we show separately the outflowing (1st and 3rd row) and inflowing (2nd and 4th row) star formation components based on radial velocity. The fraction of total
+galaxy SFR in the inflowing gas component is indicated in each case. The inflowing and outflowing components for the no-wind case are very similar in their
+contribution to the SFR, varying anywhere between ∼ 40–60% of the total SFR in any one of the components. AGN winds introduce stark differences in the
+amount and distribution of star-forming gas, with the inflowing and outflowing components contributing as much as ∼80–90% of the total SFR at different
+times.
+Alcázar et al. 2021), suggesting that this is a likely phase for strong
+AGN activity as well. We thus investigate the plausible positive and
+negative effects of AGN feedback at the galaxy’s peak of nuclear and
+star formation activity.
+2.3 Hyper-refined AGN winds
+The method to inject AGN winds at super-Lagrangian resolution in
+cosmological simulations is described in Anglés-Alcázar et al. (in
+prep.), and builds on earlier particle spawning implementations in
+idealized simulations of galaxies and massive halos (Richings &
+Faucher-Giguère 2018a; Torrey et al. 2020; Su et al. 2021). The
+BH is modelled as a collisionless particle with initial mass MBH =
+109 M⊙ and located near the centre of the main galaxy. The accretion
+rate is assumed to be constant, for simplicity, throughout the dura-
+tion of the simulation, representing a luminous quasar phase lasting
+∼ 40 Myr with the BH accreting at a fixed fraction λEdd of the Ed-
+dington rate. Stochastic swallowing of gas particles within the BH
+interaction kernel (defined to contain ∼ 256 particles) ensures mass
+conservation (e.g. Anglés-Alcázar et al. 2017a).
+Our AGN wind model is specified by the following main proper-
+ties: the mass outflow rate ˙Mw, the initial wind velocity vw, and the
+geometry of the wind. We consider that a fraction ϵk of the AGN
+bolometric luminosity (Lbol ≡ 0.1 ˙MBH c2) emerges as a fast, nuclear
+isotropic wind radially outward from the BH, with initial velocity
+vw = 30, 000 km s−1 and temperature Tw ∼ 104 K, typical of broad
+absorption line winds and ultrafast outflows (Weymann et al. 1981;
+Gibson et al. 2009; Feruglio et al. 2015; Nardini et al. 2015; Tombesi
+et al. 2015). We assume that the wind immediately interacts with the
+ambient medium, with post-shock velocity and temperature given
+by vsh = vw/4 = 7, 500 km s−1 and Tsh ≈ 1.2 × 1010 K (Faucher-
+Giguère & Quataert 2012). In practice, resolving the shock structure
+is challenging (Richings & Faucher-Giguère 2018a,b; Torrey et al.
+2020) and we model the AGN wind by creating or spawning new
+gas particles within a sphere Rw = 0.1 pc around the BH, with ini-
+tial velocity vsh and temperature Tsh consistent with post-shock con-
+ditions. Other fluid quantities are immediately recomputed for the
+wind particles after spawning, interacting hydrodynamically with
+the ISM gas in the simulation. The simulations presented here imple-
+ment a target wind particle mass of 1000 M⊙ h−1, which represents
+a factor >20 times higher mass resolution than the original simula-
+tion, and we consider discrete ejection events containing between 10
+and 100 wind particles distributed isotropically and moving radially
+outward from the BH. Particle spawning allows us to fully capture
+the propagation and impact of fast winds at higher resolution than
+normally possible with Lagrangian hydrodynamics (see also Costa
+et al. 2020), injecting feedback locally around the BH and capturing
+the wind-ISM interaction robustly regardless of gas geometry and at
+MNRAS 000, 1–16 (2023)
+
+Local positive AGN feedback in the overall negative
+5
+significantly higher resolution than nearest neighbor-based feedback
+coupling models.
+Table 1 summarizes the main properties of the simulations anal-
+ysed here. All simulations start from the same initial conditions de-
+scribed in §2.2, containing a central BH with mass MBH = 109 M⊙,
+and implementing the same post-shock wind velocity and temper-
+ature while varying the mass outflow rate
+˙Mw. We assume that
+the BH accretes at the Eddington rate (λEdd = 1), motivated by
+hyper-refinement simulations that predict quasar-like inflow rates
+at sub-pc scales for the same simulated galaxy conditions (Anglés-
+Alcázar et al. 2021). Along with the standard FIRE-2 simulation
+that excludes AGN feedback (no-wind), we investigate the impact
+of AGN-driven winds with kinetic feedback efficiencies in the range
+ϵk = 0.5–50%, which brackets a range of observational constraints
+(e.g. Cicone et al. 2014; Fiore et al. 2017; Harrison et al. 2018) and
+assumed feedback efficiencies in previous simulations (e.g. Di Mat-
+teo et al. 2005; Weinberger et al. 2017; Davé et al. 2019). The sim-
+ulation name in each feedback implementation encodes the value
+of the mass loading factor (ηk) and the kinetic feedback efficiency
+(ϵk × 100). The two simulations that we reference the most through-
+out this work are:
+• no-wind: The control simulation using standard FIRE-2 physics,
+where we model the evolution of a massive galaxy (Mstar ∼ 1011 M⊙)
+starting at z ∼ 2.28 (∆t = 0 Myr) and no AGN winds are introduced.
+The BH is still accreting at the Eddington rate, ˙MBH ∼ 22.2 M⊙ yr−1.
+• AGN-wind: Fiducial simulation where AGN winds are turned on
+at ∆t = 0 Myr with the same initial conditions as the no-wind case.
+We consider a luminous quasar phase with bolometric luminosity
+Lbol = 1.26 × 1047 erg s−1, driving a wind with kinetic efficiency
+ϵk = 0.1 and mass loading factor ηk ≡ ˙Mw/ ˙MBH = 2, corresponding
+to a mass outflow rate in winds ˙Mw = 44.4 M⊙ yr−1.
+The times mentioned in this work are relative to the start of the wind
+phase at t0, with ∆t referring to the time that has passed since then
+as ∆t ≡ t − t0.
+3 OVERVIEW OF SIMULATIONS
+Figure 1 shows the projected star formation rate surface density for
+two different simulations of the same massive star-forming galaxy
+(Mstar ∼ 1011 M⊙, SFR ∼ 300 M⊙ yr−1) at z ∼ 2.28, one with
+no AGN feedback (no-wind), and the other with feedback (AGN-
+wind), at varying snapshots in time. In the no-wind case (top two
+rows), the central BH accretes gas at the Eddington limit but we
+neglect any AGN feedback effects, with stellar feedback solely re-
+sponsible for regulating star formation. At the beginning of the sim-
+ulated period, the galaxy resembles a turbulent, clumpy, kpc-scale
+disc with significant amounts of star formation occurring along frac-
+tured spiral arms and the denser nuclear region. As time proceeds,
+the dense, star-forming gas reservoir continues to be replenished by
+infalling gas across scales and becomes more concentrated toward
+the centre, forming an ultra-compact, nuclear star-forming disc after
+∆t ∼ 25 Myr of evolution. From the first snapshot (∆t = 0.2 Myr) to
+the last (∆t = 35 Myr), the SFR in the central 1 kpc region increases
+from ∼ 400 M⊙ yr−1 to ∼ 640 M⊙ yr−1.
+In the AGN-wind simulation (bottom two rows), we inject radial
+shells of high resolution particles to represent the AGN wind. In the
+first ∆t = 1 Myr of evolution there is already a noticeable effect on
+the galaxy. The wind has ejected star-forming gas from the central
+50 pc as well as cleared out other areas throughout the disc that had
+high SFR (dark green regions) in the no-wind case, that now have
+lower amounts of star formation (lighter green regions) or none at
+all. Winds propagate more efficiently along paths of least resistance,
+preferentially along the rotation axis of the galaxy, but can nonethe-
+less penetrate through low-density ISM channels across the galaxy
+disc. After 5 Myr, the morphology of the star-forming gas becomes
+drastically different, with AGN winds opening up a growing, central
+cavity surrounded by strips of gas, in contrast to the ultra-compact
+star-forming disc in the absence of AGN feedback. Not only is the
+morphology different than in the no-wind case, but the amount of
+star formation is drastically different as well. By ∆t = 0.2 Myr, the
+AGN winds have already had a clear impact on the nuclear SFR,
+with ∼ 230 M⊙ yr−1 within 1 kpc, a factor of ∼ 0.5 less than in the
+no-wind simulation. At a later time, 25 Myr, the total SFR in the
+AGN-wind case reaches ∼ 19 M⊙ yr−1, constituting a decrease by
+a factor of ∼ 33 from the no-wind case, an overwhelmingly nega-
+tive effect. By the end of the simulation, (∆t = 35 Myr), the total
+SFR within 1 kpc in the no-wind case is ∼ 640 M⊙ yr−1, but only
+∼ 0.4 M⊙ yr−1 in the AGN-wind case, where most of the dense star-
+forming gas has been evacuated by the winds. In the central 250 pc
+region, the SFR increases from ∼ 230 M⊙ yr−1 (∆t = 0.2 Myr) to
+∼ 600 M⊙ yr−1 (∆t = 35 Myr) in the no-wind simulation, while in
+the AGN-wind case it goes from ∼ 90 M⊙ yr−1 (∆t = 0.2 Myr) to
+∼ 0.003 M⊙ yr−1 (∆t = 35 Myr). While the global SFR increases by
+40% during the no-wind simulation, the SFR in the AGN-wind case
+decreases by a factor of ∼ 3000.
+Figure 2 shows the 2D distribution of star-forming gas in the ra-
+dial velocity–radial distance plane for the no-wind (top) and AGN-
+wind (bottom) simulations. For each case, we show separately the
+inflowing (vr < 0) and outflowing (vr > 0) components of the star-
+forming gas, which are defined relative to the radial velocity (vr) with
+respect to the BH. We thus refer to inflows and outflows as radially
+inward and outward components without requiring a minimum abso-
+lute velocity. Some of the morphological features that were present
+in Figure 1 are identifiable within the 2D distribution for the no-
+wind simulation, such as the vertical stripes corresponding to spi-
+ral arms. As time progresses, the star-forming gas becomes more
+centrally concentrated, which is seen here as the distribution shift-
+ing to the left. Aside from the increase in total SFR, there is an in-
+crease in the net radial velocity with time, reaching ∼2,000 km/s on
+10–100 pc scales for both the outflowing and inflowing components
+(∆t = 20−25 Myr). In the no-wind simulation, the two radial velocity
+components show similar distributions and contribute approximately
+similar amounts to the global SFR, with a slight predominance of the
+inflow component at later times. In the absence of coherent, large
+scale inflow/outflow conditions dominating the gas’ dynamics, the
+roughly symmetric radial velocity structure is the result of highly
+turbulent motions in the ISM. The intense star formation in the nu-
+clear region at high surface densities (ΣSFR > 1000 M⊙ yr−1 kpc−2)
+rapidly deepens the potential well, resulting in the increased maxi-
+mum radial velocities observed.
+In the AGN-wind simulation, the radial distribution of star-
+forming gas clearly shows the impact of AGN feedback, with the
+size of the central cavity growing with time on average (bottom
+rows). Despite the injection of positive radial momentum, there is
+no significant sign of SFR enhancement in the outflowing compo-
+nent. During the first ∼10 Myr of evolution, the inflow and outflow
+components contribute roughly equally to the total SFR, similar to
+the no-wind case. However, the inflow component tends to dominate
+at later times, when the impact of winds becomes more dramatic.
+This suggests that gas radially accelerated by the winds has gener-
+ally lower probability of forming stars. One interesting exception is
+MNRAS 000, 1–16 (2023)
+
+6
+J. Mercedes-Feliz et al.
+3.0
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+log10(R) [kpc]
+6
+4
+2
+0
+2
+4
+6
+log10(
+SFR) [M  yr
+1 kpc
+2]
+No-wind
+No-wind
+No-wind
+No-wind
+No-wind
+No-wind
+No-wind
+3.0
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+log10(R) [kpc]
+AGN-wind
+AGN-wind
+AGN-wind
+AGN-wind
+AGN-wind
+AGN-wind
+AGN-wind
+0.2 Myr
+5.0 Myr
+10.0 Myr
+15.0 Myr
+20.0 Myr
+25.0 Myr
+35.0 Myr
+Figure 3. Radial profile of star formation rate surface density (ΣSFR) for the no-wind (left; blue) and AGN-wind (right; orange) simulations. Time evolution is
+indicated by the saturation of the coloured lines. The high nuclear ΣSFR in the no-wind simulation continues to increase with time while AGN winds create a
+central cavity and suppress the overall SFR.
+the outflowing component at ∆t = 25 Myr since the start of the AGN
+wind phase, which contains a considerable amount of star formation
+(16 M⊙ yr−1, corresponding to >80% of the total SFR) in a localized
+region at 300 pc from the centre and with radial velocity as high as
+800 km s−1 on average.
+4 NEGATIVE GLOBAL IMPACT OF AGN FEEDBACK
+Figure 3 shows the azimuthally-averaged radial profile of the star
+formation rate surface density for the no-wind (left) and AGN-
+wind (right) simulations, for various times. We use cylindrical ra-
+dial bins that are defined relative to the angular momentum rotation
+axis of star-forming gas within 2 kpc (since the majority of the star-
+forming gas is contained within this scale) of width ∆ log10(R) =
+0.1 dex to calculate the SFR per unit area in each bin (ΣSFR). For clar-
+ity, we smooth the resulting radial distributions by applying a run-
+ning average considering the two nearest neighbor bins. At the start
+of the simulation in the no-wind case, the star-forming disc reaches
+ΣSFR ∼ 1000 M⊙ yr−1 kpc−2 in the nuclear region (10–100 pc), while
+maintaining ΣSFR ∼ 10 M⊙ yr−1 kpc−2 on kpc scales. In the ab-
+sence of AGN winds, the nuclear gravitational potential deepens
+significantly and the radial distribution of star-forming gas steepens
+strongly over the 35 Myr period, reaching extreme surface densities,
+up to ΣSFR ∼ 104 M⊙ yr−1 kpc−2 in the inner pc, and containing most
+star formation within ∼200 pc.
+In contrast to the no-wind simulation, the formation of the cen-
+tral cavity is clearly visible in the radial SFR profile of the AGN-
+wind simulation. The size of the cavity increases with time, on av-
+erage, as winds continue to inject energy and momentum into the
+surrounding gas. However, the persistent, non-isotropic infall of gas
+onto the galaxy and the inefficient coupling of winds with low-
+subtended area gas structures drives fluctuations in the size of the
+cavity, with dense gas clumps sometimes able to penetrate the inner
+cavity down to <10 pc scales. Besides the cavity opening, the ampli-
+tude of the ΣSFR profile also decreases with time across radial scales,
+emphasizing the overall negative impact of AGN winds on the star
+formation properties of the host galaxy.
+Figure 4 shows the total stellar mass enclosed in the central 2 kpc
+of the galaxy as a function of time since the start of the AGN
+feedback phase at ∆t = 0, excluding stars formed at earlier times
+(solid lines). In the no-wind simulation (blue), the extreme SFR sur-
+face densities reached lead to the formation of ∼ 2 × 1010 M⊙ of
+stars during only 35 Myr. In contrast, the overall reduction in SFR
+in the AGN-wind simulation (orange) yields the formation of only
+∼ 3 × 109 M⊙ of stars during the initial 10 Myr, with a subsequent
+∼60% decrease in enclosed stellar mass within 2 kpc owing to the
+expansion of the stellar component driven by the expulsion of the
+nuclear gas reservoir as well as stellar mass return to the ISM and
+significantly lower SFR at later times.
+In order to understand the effect of AGN winds, we identify
+all gas elements within 2 kpc that have non-zero SFR at t = t0
+and track them in time by means of their unique identifiers, which
+are preserved when gas elements are converted into star particles
+(e.g. Anglés-Alcázar et al. 2017b). The horizontal dash-dotted line
+(black) indicates the corresponding total initial amount of star-
+forming gas available at t = t0. The blue and orange dashed lines
+represent the amount of stellar mass formed from this selected ini-
+tial star-forming gas reservoir in the no-wind and AGN-wind simula-
+tions, respectively, while the leftover gas mass is shown as the dotted
+lines.
+In the no-wind simulation, almost all of the initial star-forming gas
+MNRAS 000, 1–16 (2023)
+
+Local positive AGN feedback in the overall negative
+7
+0
+5
+10
+15
+20
+25
+30
+35
+t = t
+t0 [Myr]
+7.5
+8.0
+8.5
+9.0
+9.5
+10.0
+log10(Mass) [M ]
+R < 2 kpc
+M (t)
+M (t0)
+Stars from SF gas at t0
+Leftover SF gas from t0
+No-wind
+AGN-wind
+Figure 4. Stellar mass growth within the central 2 kpc as a function of time
+for the no-wind (blue) and AGN-wind (orange) simulations. Solid lines rep-
+resent the total mass of stars formed since t0, i.e. excluding any pre-existing
+stars. The horizontal dash-dotted line (black) represents the initial total mass
+of star-forming gas available at the start of the simulations (t0), the dashed
+lines indicate the stellar mass formed from this gas, and the dotted lines show
+the remaining gas mass from originally star-forming gas. The suppression of
+stellar mass growth by AGN winds over ∼35 Myr is driven primarily by a
+reduction in the amount of new gas that can become star-forming as opposed
+to by directly ejecting pre-existing star-forming gas.
+is converted into stars after 35 Myr, corresponding to ∼ 109 M⊙3.
+In contrast, in the AGN-wind simulation, only ∼40% of the origi-
+nal star-forming gas is converted into stars (∼ 6 × 108 M⊙), which
+demonstrates a direct, negative impact of AGN winds on the pre-
+existing star-forming gas. However, the total amount of stellar mass
+formed in the no-wind case is more than one order of magnitude
+larger than the mass available in the initial star-forming gas reser-
+voir, indicating that the majority of stars form from additional gas
+becoming star-forming over time. The dominant negative effect of
+AGN winds over the 35 Myr period is thus to prevent the replen-
+ishment of the star-forming gas reservoir, with the direct ejection of
+pre-existing star-forming gas playing a lesser role.
+5 SIGNATURES OF (LOCAL) POSITIVE AGN FEEDBACK
+5.1 Spatial anti-correlation of winds and star-forming regions
+Figure 5 shows the star formation rate surface density as a function
+of radial distance (left panel), comparing the no-wind and AGN-
+wind simulations at time ∆t = 20 Myr. This highlights again the
+overall negative impact of AGN winds, creating a central cavity de-
+void of star-forming gas, in stark contrast to the extreme ΣSFR values
+reached in the absence of winds. However, this also shows that ΣSFR
+can be larger in the presence of AGN winds in localized regions un-
+der certain conditions. In this case, the azimuthally-averaged ΣSFR at
+3 There is ∼20% of “missing” mass due to stellar mass loss as well as BH
+accretion.
+a radial distance of ∼150–250 pc is larger in the AGN-wind simula-
+tion compared to the no-wind simulation (grey shaded area), which
+represents a plausible indication of local positive AGN feedback.
+The same effect is illustrated in the right panel, where we show
+the projected, face-on spatial distribution of star-forming gas, over-
+laying the no-wind (grey scale) and AGN-wind (colour scale) simu-
+lations. Most star formation in the presence of AGN winds occurs in
+dense gas clumps located at >100 pc, outside of the wind-dominated
+region. This spatial anti-correlation of winds and high star-forming
+regions suggests that the accumulation and compression of gas by
+AGN feedback in certain regions can enhance, rather than suppress,
+local star formation.
+5.2 Enhanced star formation efficiency by AGN wind
+compression
+Figure 6 investigates in more detail the local triggering of star forma-
+tion by AGN winds, comparing the radial profiles of star formation
+efficiency (SFE) and molecular gas mass surface density (Σmol) at
+times ∆t = 10 Myr and ∆t = 20 Myr for the no-wind and AGN-
+wind simulations (left panels). We compute the SFE of each gas el-
+ement as the SFR divided by mass, SFE ≡ SFR/Mgas, which is then
+averaged (mass-weighted) over all gas elements in cylindrical radial
+bins. For Σmol, we assume that gas elements with Hydrogen number
+density nH > 1000 cm−3 have a molecular fraction near unity. We
+also show face-on projections of the spatial distribution of SFE for
+the AGN-wind simulation (right panels).
+In both simulations we see a consistent trend of higher SFE with
+higher molecular gas surface density, as expected, but they do not
+follow a one-to-one relation because the SFR of a given gas clump
+depends on its free-fall time, virial parameter, and other factors
+(Hopkins et al. 2018). In the no-wind simulation, SFE and Σmol both
+increase by more than three orders of magnitude from kpc to pc
+scales at ∆t = 10 Myr, and by almost five orders of magnitude at
+∆t = 20 Myr, since gas had more time to accumulate in the cen-
+tre, becoming more centrally concentrated. In the presence of AGN
+feedback, SFE and Σmol also tend to increase with decreasing radial
+distance, but both drop very rapidly inside of the central cavity evac-
+uated by the AGN winds.
+A local enhancement of SFR in the presence of AGN winds rel-
+ative to the no-wind case, such as seen at ∆t = 20 Myr (Figure 5),
+could be due to an increase in the amount of dense gas available
+in a given region and/or an increase in the efficiency of converting
+gas into stars (Moreno et al. 2021). The bottom left panel of Figure 6
+shows that, at ∆t = 20 Myr, the increase in local SFR surface density
+in the range ∼150–250 pc under the presence of AGN winds coin-
+cides with the SFE being up to one order of magnitude higher in the
+AGN-wind simulation compared to the no-wind case. In the same ra-
+dial range, however, the amount of fuel for star formation (molecular
+gas) is similar both in the presence and absence of AGN winds, sug-
+gesting that the increase in local SFR is more efficiency-driven than
+fuel-driven in this case. We find similar results when considering the
+SFE per free-fall time (Fig. 6.
+The earlier physical conditions at ∆t = 10 Myr (Figure 6, top
+panels) also represent an interesting example of increased SFE lo-
+cally (R ∼ 100 pc) in the presence of AGN winds despite the overall
+suppression of global star formation relative to the no-wind simula-
+tion. The top right panel of Fig 6 shows that gas clumps with high
+SFE clearly form along the edge of the cavity, where the incident
+AGN winds interact with the ISM. In this case, they also have higher
+molecular gas surface density compared to the no-wind simulation.
+Overall, these results suggest that AGN winds can trigger star for-
+MNRAS 000, 1–16 (2023)
+
+8
+J. Mercedes-Feliz et al.
+3.0
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+log10(R) [kpc]
+3
+2
+1
+0
+1
+2
+3
+4
+5
+6
+log10(
+SFR) [M  yr
+1 kpc
+2]
+t = 20.0 Myr
+No-wind
+AGN-wind
+200 pc
+log10(
+SFR) [M  yr
+1 kpc
+2]
+log10(
+SFR) [M  yr
+1 kpc
+2]
+0
+1
+2
+3
+4
+5
+Figure 5. Left: Radial profile of the total star formation rate surface density (ΣSFR) corresponding to ∆t = 20 Myr since the start of the AGN feedback phase.
+The blue line represents the no-wind case while orange is the AGN-wind case. The vertical grey band indicates the region where ΣSFR is higher in the AGN-
+wind simulation compared to the no-wind case. Right: Projected star formation surface density map with the no-wind (grey scale) and AGN-wind (colour scale)
+simulations superimposed. The radial annulus corresponds to the vertical grey band in the left panel. Despite the overall negative effect of BH feedback, ΣSFR
+can reach higher values in certain regions under the presence of AGN winds, suggesting local positive feedback effects.
+mation and enhance the local star formation efficiency locally by
+compressing the ISM, while having a global negative effect.
+5.3 Contribution of outflowing gas to star formation
+Figure 7 shows the radial profiles of the SFR surface density at
+∆t = 15 Myr and ∆t = 25 Myr (left), separating the distribution into
+inflowing and outflowing components based on the radial velocity of
+gas, as in Figure 2. We also show the projected, face-on distributions
+of ΣSFR in the right panels, overlaying the no-wind (grey scale) and
+AGN-wind (colour scale) simulations.
+In the no-wind simulation at ∆t = 15 Myr, a single spiral arm sur-
+rounds the ultra-compact nuclear disc forming on 100 pc scales. As
+shown above, introducing AGN feedback changes the morphology
+of the galaxy, forming a slightly off-centred cavity ∼400 pc wide by
+evacuating the nuclear gas disc and pushing outward the dominant
+spiral arm. The inset figure in the ΣSFR map indicates the spatial dis-
+tribution of the inflowing (blue) and outflowing (red) star-forming
+gas, showing that the region near the cavity edge is dominated by
+the outflow component. Interestingly, the inflow component reaches
+inside of the cavity down to the inner ∼10 pc, previously devoid of
+dense gas, despite the continuous injection of winds pushing ma-
+terial outwards. At ∆t = 25 Myr, the centre of the cavity has been
+cleared out again by the winds, but the overall size of the cavity has
+decreased to ∼100 pc in diameter, owing to the large inflow of gas
+occurring during the previous ∼10 Myr. At this time, the dense, ring-
+like gas structure in the AGN-wind simulation is outflow-dominated
+and on the way to expand again, in contrast to the ultra-compact nu-
+clear spiral prevalent in the no-wind simulation. This illustrates the
+complex interaction between AGN-driven winds and infalling gas in
+the strong nuclear gravitational potential of a massive galaxy.
+In the no-wind simulation, the inflow and outflow components of
+star-forming gas exhibit roughly similar radial distributions, indica-
+tive of turbulent motions (see also Figure 2), with the overall contri-
+bution of inflowing material to global star formation being slightly
+predominant over the outflow component for both ∆t = 15 Myr and
+∆t = 25 Myr. In the AGN-wind simulation, however, we identify a
+decoupling of dynamical components, with certain regions clearly
+dominated by outflowing material. In the radial range indicated by
+the vertical grey bands, the outflow component reaches ΣSFR values
+between one (∆t = 25 Myr) and two (∆t = 15 Myr) orders of mag-
+nitude higher than the inflow component, coinciding with gas struc-
+tures along the edge of the cavity compressed by the AGN winds.
+Thus, even in cases such as these, where AGN winds suppress ΣSFR
+at all radii relative to the no-wind case, the higher fractional contri-
+bution of outflowing material to star formation can be indicative of
+local AGN feedback triggering of star formation, as star-forming gas
+is preferentially entrained within the outflowing medium.
+5.4 Spatial and temporal shift in star formation
+Since all simulations start from the same initial conditions (§2.2), it
+is possible to track the evolution of the same Lagrangian mass ele-
+ments (either gas or stars) across simulations using their unique par-
+ticle identifiers. Star formation is so efficient in the no-wind case that
+the majority of the gas particles that turned into stars in the AGN-
+wind case also formed stars in the no-wind simulation, which allows
+us to look further into the effect of AGN feedback on the stars that
+form in both cases. Figure 8 quantifies the difference in the forma-
+tion time and radial distance for stars formed in the AGN-wind sim-
+ulation relative to the stars formed out of the same gas elements in
+the no-wind simulation. Here, we show the two-dimensional distri-
+MNRAS 000, 1–16 (2023)
+
+Local positive AGN feedback in the overall negative
+9
+3.0
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+log10(R) [kpc]
+9.0
+8.5
+8.0
+7.5
+7.0
+6.5
+6.0
+5.5
+5.0
+log10(SFE) [yr
+1]
+t = 10.0 Myr
+3
+4
+5
+6
+7
+8
+9
+10
+log10(
+mol) [M  kpc
+2]
+3
+4
+5
+6
+7
+8
+9
+10
+log10(
+mol) [M  kpc
+2]
+No-wind
+AGN-wind
+SFE
+mol
+log10(SFE) [yr
+1]
+200 pc
+9.5
+8.5
+7.5
+6.5
+5.5
+3.0
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+log10(R) [kpc]
+9.0
+8.5
+8.0
+7.5
+7.0
+6.5
+6.0
+5.5
+5.0
+log10(SFE) [yr
+1]
+t = 20.0 Myr
+3
+4
+5
+6
+7
+8
+9
+10
+log10(
+mol) [M  kpc
+2]
+3
+4
+5
+6
+7
+8
+9
+10
+log10(
+mol) [M  kpc
+2]
+No-wind
+AGN-wind
+SFE
+mol
+log10(SFE) [yr
+1]
+200 pc
+9.5
+8.5
+7.5
+6.5
+5.5
+Figure 6. Left: Radial profile of the star formation efficiency (SFE; solid lines) and the total molecular gas surface density (Σmol; dashed lines) at time
+∆t = 10 Myr (top) and ∆t = 20 Myr (bottom) since the beginning of the AGN feedback phase for the no-wind (blue) and AGN-wind (orange) simulations. The
+vertical grey band indicates the region where SFE is higher in the AGN-wind simulation compared to the no-wind case. Right: Spatial SFE distribution for a
+face-on projection of the AGN-wind case. The radial annulus corresponds to the vertical grey band in the left panel. The star formation efficiency increases
+along the edge of the central cavity relative to the no-wind simulation, suggesting that compression of ISM by AGN winds can trigger star formation locally.
+bution for the differences in formation time and distance for stars
+that formed during the 30 Myr since the start of the AGN feedback
+phase (defined for the AGN-wind simulation). This allows us to in-
+vestigate whether stars that form in the presence of AGN winds do
+so at similar times/distances or not compared to the same stars in the
+absence of winds. For example, the top right quadrant corresponds
+to stars forming later in time and farther from the centre in the AGN-
+wind run, while the lower left quadrant represents stars forming ear-
+lier and closer than in the no-wind case.
+The distribution in the difference of radial distances for stars that
+formed under the presence of AGN winds shows that ∼80% of stars
+preferentially formed at a further distance compared to their no-
+wind counterpart. This may constitute indication of negative feed-
+back, even for stars that manage to form in the presence of AGN
+feedback. When investigating the difference in formation times, we
+find some indication of earlier conversion of gas into stars for stars
+that form further out in the AGN-wind simulation (with AGN winds
+pushing ISM gas radially outward but locally triggering faster star
+formation) while preferentially delayed star formation for stars that
+form closer to the BH. The complex interaction of AGN winds and
+ISM gas drives a weak anti-correlation between the difference in for-
+MNRAS 000, 1–16 (2023)
+
+10
+J. Mercedes-Feliz et al.
+3.0
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+log10(R) [kpc]
+6
+4
+2
+0
+2
+4
+6
+log10(
+SFR) [M  yr
+1 kpc
+2]
+t = 15.0 Myr
+No-wind
+AGN-wind
+Inflow
+Outflow
+log10(
+SFR) [M  yr
+1 kpc
+2]
+log10(
+SFR) [M  yr
+1 kpc
+2]
+0
+1
+2
+3
+4
+5
+Inflow
+Outflow
+3.0
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+log10(R) [kpc]
+6
+4
+2
+0
+2
+4
+6
+log10(
+SFR) [M  yr
+1 kpc
+2]
+t = 25.0 Myr
+No-wind
+AGN-wind
+Inflow
+Outflow
+log10(
+SFR) [M  yr
+1 kpc
+2]
+log10(
+SFR) [M  yr
+1 kpc
+2]
+0
+1
+2
+3
+4
+5
+Inflow
+Outflow
+Figure 7. Left: Radial profile of the total star formation rate surface density (ΣSFR) for the inflowing material (solid lines) and outflowing material (dashed
+lines) at time ∆t = 15 Myr (top) and ∆t = 25 Myr (bottom) since the beginning of the AGN feedback phase for the no-wind (blue) and AGN-wind (orange)
+simulations. Right: Spatial star formation surface density distribution for a face-on projection with the no-wind (grey scale) and AGN-wind (colour scale)
+simulations superimposed. Inset: A closer look at the central 250 pc (top) and 200 pc (bottom) region, decomposing the AGN-wind map into inflowing (blue)
+and outflowing (red) components. The vertical grey band (left) and corresponding radial annulus (right) indicate the region where the outflowing component
+dominates over the inflowing component for the AGN-wind simulation. The increased local fractional contribution of outflowing gas to star formation is a
+plausible signature of positive AGN feedback.
+mation distance and the difference in formation time relative to the
+no-wind simulation.
+Figure 9 examines the radial velocities of the same sample of
+stars as in Figure 8, where we now show separate distributions for
+stars that formed in the AGN-wind simulation during three differ-
+ent time intervals: 0 ≤ tform < 10 Myr, 10 ≤ tform < 20 Myr, and
+20 ≤ tform < 30 Myr. In the no-wind simulation, the stellar ra-
+dial velocity distributions are roughly symmetric and spread up to
+±1000 km s−1, owing to the strong deepening of the nuclear grav-
+itational potential in the absence of AGN feedback, with only mi-
+nor differences depending on the stellar formation time. For com-
+parison, the escape velocity at 1 kpc increases from ∼620 km s−1 at
+MNRAS 000, 1–16 (2023)
+
+Local positive AGN feedback in the overall negative
+11
+1.00
+0.75
+0.50
+0.25 0.00
+0.25
+0.50
+0.75
+1.00
+ Radial Distance [kpc]
+40
+30
+20
+10
+0
+10
+20
+30
+40
+ Formation Time [Myr]
+14.3%
+32.5%
+46.7%
+6.5%
+AGN-wind
+No-wind
+3-
+2-
+1-
+0
+1
+2
+3
+log10(Counts)
+Figure 8. Difference in formation time and radial distance for stars formed
+in the AGN-wind simulation relative to stars formed out of the same gas
+elements in the no-wind simulation. The two-dimensional distribution (and
+corresponding one-dimensional histograms) are shown for stars that formed
+in the AGN-wind simulation during : 0 �� tform < 30 Myr. The radial distance
+to the centre is always measured at ∆t = 30 Myr. The contour lines high-
+light the regions that contains 1-σ, 2-σ, and 3-σ of the distribution. Dashed
+lines separate the distributions into four quadrants centred at [0,0], with the
+corresponding fraction of stellar mass indicated in each quadrant. Stars form
+preferentially farther out under the presence of AGN winds compared to the
+no-wind simulation (right quadrants), with a weak correlation between dif-
+ference in formation time and distance.
+∆t = 5 Myr to ∼700 km s−1 at ∆t = 25 Myr, while at 100 pc the
+escape velocity increases from ∼750 km s−1 to ∼1150 km s−1 in the
+same time interval. In the absence of strong AGN or stellar-feedback
+driven winds that could potentially accelerate star-forming clumps,
+the large stellar velocities reached in the no-wind simulation can thus
+be easily explained by orbital dynamics in the ultra-dense nuclear
+stellar potential (Anglés-Alcázar et al. 2017c; Wellons et al. 2020;
+Parsotan et al. 2021).
+In contrast, we find that new stars in the AGN-wind simulation are
+kinematically different in addition to forming further out compared
+to the no-wind case, exhibiting lower radial velocities than the same
+stars in the no-wind simulation. This is particularly the case for stars
+forming in the first ∼ 20 Myr, where ∼90% end up with radial veloc-
+ities within ±250 km s−1, owing to the lower gravitational potential.
+In this case, stars that form in the last ∼10 Myr in the presence of
+winds show a broader velocity distribution compared to stars formed
+earlier, extending up to ±500 km s−1 possibly due to the more coher-
+ent nature of star-forming structures dominated by either inflowing
+or outflowing material at higher velocities (Fig. 2). In any case, stel-
+lar radial velocities are a better reflection of the depth of the gravita-
+tional potential than of the velocity of AGN-driven winds.
+6 DEPENDENCE ON AGN WIND EFFICIENCY
+Figure 10 shows the star formation rate surface density (ΣSFR) as a
+function of radial distance, similar to Figure 5 but at ∆t = 10 Myr, for
+Radial Velocity [km s
+1]
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+Fraction of Stars
+No-wind
+1000
+750
+500
+250
+0
+250
+500
+750
+1000
+Radial Velocity [km s
+1]
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+Fraction of Stars
+AGN-wind
+Stellar Formation Time
+0
+ tform < 10 Myr
+10
+ tform < 20 Myr
+20
+ tform < 30 Myr
+Figure 9. Radial velocity distribution for stars that formed in the AGN-
+wind simulation within ∼ 30 Myr (bottom panel), and the same star particles
+tracked in the no-wind simulation (top panel). Stars are separated according
+to their formation time in the AGN-wind simulation, as in Figure 8, with
+each colour corresponding to stars that formed in a given time interval as
+indicated. The radial velocity is always measured at ∆t = 30 Myr.
+the no-wind and AGN-wind simulations, as well as four other sim-
+ulations that vary in AGN feedback strength (see Table 1). At first
+glance, the hierarchy in feedback strength, from weakest (ϵk = 0.5%;
+green) to strongest (ϵk = 50%; red), can be clearly seen in the
+increasing suppression of the star formation rate surface density
+distribution relative to the no-wind case. The weakest AGN winds
+can barely open a central cavity of size ∼ 25 pc during the first
+10 Myr, while the strongest winds very quickly evacuate the entire
+star-forming gas reservoir within the central kpc. Aside from seeing
+how ΣSFR changes with different feedback efficiencies, we can also
+identify examples of local positive feedback. AGN winds in sim-
+ulation m0.1e0.5 can only suppress star formation within a small
+cavity, but the compression of gas in the edge of the cavity triggers
+star formation reaching a factor of ten higher ΣSFR than in the no-
+wind simulation in the same radial range.
+The top panel of Figure 11 shows the stellar mass enclosed within
+R < 2 kpc as a function of time, excluding the mass found already
+in stars at the beginning of the AGN feedback phase at t0, as in Fig-
+ure 4 but now comparing simulations with different AGN feedback
+strength. Similarly, the middle panel shows the stellar mass formed
+out of the initial star-forming gas reservoir at t0 for each simula-
+tion, and the bottom panel shows the corresponding leftover star-
+forming gas. The m0.1e0.5 simulation shows very similar stellar
+mass growth as the no-wind case over time, indicating that AGN
+winds with kinetic feedback efficiency ϵk = 0.5% and energy injec-
+tion rate ˙Ew ∼ 6.29 × 1044 erg s−1 can only barely affect the star-
+forming gas reservoir of this massive galaxy. Increasing the AGN
+feedback strength by a factor of ten (ϵk = 5%) has a clear negative
+effect in the overall stellar mass growth, suppressing the conversion
+of initially star-forming gas into stars and, more importantly, reduc-
+ing the amount of new gas that can become star-forming, as seen
+in Figure 4 for our fiducial AGN-wind simulation. In the strongest
+feedback cases (m4e20 and m10e50), the total stellar mass enclosed
+within 2 kpc increases initially by ∆M⋆ ∼ 109 M⊙ (during the first
+MNRAS 000, 1–16 (2023)
+
+12
+J. Mercedes-Feliz et al.
+3.0
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+log10(R) [kpc]
+4
+2
+0
+2
+4
+6
+log10(
+SFR) [M  yr
+1 kpc
+2]
+t = 10.0 Myr
+No-wind
+m0.1e0.5
+m1e5
+AGN-wind
+m4e20
+m10e50
+Figure 10. Radial profile of the total star formation rate surface density
+(ΣSFR) corresponding to ∆t = 10 Myr since the start of the AGN phase, for
+various simulations with different AGN feedback strength.
+∼5 Myr) but then quickly decreases owing to the very strong sup-
+pression of subsequent star formation and the change in the gravita-
+tional potential due to the massive evacuation of gas from the galaxy
+driving the expansion of the stellar component.
+7 DISCUSSION
+Our simulations suggest that powerful AGN winds have a global
+negative impact on the stellar mass growth of massive galaxies near
+their peak of star formation activity (Anglés-Alcázar et al., in prep.),
+in broad agreement with many previous cosmological simulations
+where AGN feedback prescriptions are calibrated to help regulate
+star formation in galaxies at the high mass end (Di Matteo et al.
+2005; Baldry et al. 2006; Bower et al. 2006; Dubois et al. 2012;
+Somerville & Davé 2015; Choi et al. 2012, 2018; Davé et al. 2019;
+Wellons et al. 2022). In the absence of AGN feedback, an ultra dense
+central starburst quickly develops at the time at which stellar feed-
+back no longer becomes efficient enough to drive large-scale galactic
+winds (Anglés-Alcázar et al. 2017c; Stern et al. 2021; Pandya et al.
+2021; Byrne et al. 2022) and the simulated galaxy becomes overmas-
+sive and overcompact relative to observations (Wellons et al. 2020;
+Parsotan et al. 2021). Our results indicate that sufficiently strong
+AGN winds (as would be produced by a MBH with MBH = 109 M⊙
+accreting at the Eddington rate with kinetic feedback efficiency
+ϵk = 5%) can shut down star formation at this critical time, pro-
+ducing more realistic galaxy sizes and central stellar densities (see
+Cochrane et al. in prep.). However, a factor of ten reduction in either
+MBH, ϵk, accretion rate relative to Eddington, or a combination of
+them (since these parameters are degenerate) can dramatically de-
+crease the impact of AGN-driven winds, suggesting that other feed-
+back channels, such as radiation pressure or cosmic rays (not in-
+cluded here) may be required to regulate massive galaxies (Costa
+et al. 2018a,b; Choi et al. 2018; Wellons et al. 2022).
+AGN winds create a central cavity devoid of star-forming gas,
+as seen in previous studies (Gabor & Bournaud 2014; Curtis & Si-
+jacki 2016; Hopkins et al. 2016; Richings & Faucher-Giguère 2018a;
+Costa et al. 2020; Torrey et al. 2020), but can also penetrate into the
+galaxy disc to some extent, reducing the star formation rate surface
+density on all scales. The coupling efficiency of AGN winds and
+the surrounding ISM decreases as the size of the cavity increases,
+with a larger fraction of input wind energy escaping along the po-
+lar direction (Torrey et al. 2020, Anglés-Alcázar et al. in prep.). The
+large, sustained gas inflow rate onto the galaxy yields a complex
+interaction between AGN winds in infalling structures, with dense
+star-forming clumps able to penetrate the cavity in some cases, and
+the geometric coupling efficiency of winds and ISM gas changing
+with time accordingly. Our results show that AGN winds reduce the
+amount of stars formed out of the initial star-forming gas reservoir
+but, more importantly, AGN winds are preventing new gas from be-
+coming star-forming throughout the simulation.
+The impact of stellar and/or AGN feedback in galaxies has been
+generically considered as either ejective, removing gas directly from
+the ISM, or preventive, stopping gas from accreting into the ISM in
+the first place (Somerville & Davé 2015). Previous studies suggest
+that preventive AGN feedback is crucial in dwarf galaxies, where
+galactic winds reduce the accretion rate onto the galaxy (Muratov
+et al. 2015; Hirschmann et al. 2016; Anglés-Alcázar et al. 2017b;
+Hafen et al. 2019, 2020; Mitchell et al. 2020; Tollet et al. 2019;
+Pandya et al. 2020), and massive halos, where “radio-mode” or
+“jet-mode” AGN feedback has been proposed to prevent hot halo
+gas from cooling (Bower et al. 2006; Gabor & Davé 2015; Wein-
+berger et al. 2017; Davé et al. 2019; Su et al. 2021), while ejective
+feedback may be more prominent in gas-rich star-forming galaxies
+(Somerville & Davé 2015). Our analysis tracking the source of gas
+responsible for star formation in each simulation suggests that AGN
+winds are acting as both modes of negative feedback simultaneously,
+ejective and preventive (see also Grand et al. 2017; Irodotou et al.
+2022). Interestingly, preventive AGN-wind feedback (in the sense of
+preventing the replenishment of the star-forming gas reservoir) ap-
+pears to be far more important than ejective feedback for suppress-
+ing the stellar mass growth of massive star-forming galaxies at their
+peak of activity (Figures 4 & 11).
+Our simulations show that AGN winds can also trigger star forma-
+tion locally by compressing gas along the edge of the cavity, increas-
+ing the gas density and star formation efficiency locally compared
+to the same region in the simulation without AGN winds. This posi-
+tive AGN feedback effect by gas compression is in qualitative agree-
+ment with previous analytic models and idealized simulations (e.g.,
+Silk 2005; Gaibler et al. 2012; Ishibashi & Fabian 2012; Zubovas &
+Nayakshin 2012; Silk 2013; Zubovas et al. 2013; Nayakshin 2014;
+Bieri et al. 2015, 2016; Dugan et al. 2017; Zubovas & Bourne 2017).
+However, while many previous studies have argued that global pos-
+itive AGN feedback plays a key role, and can even be the domi-
+nant star formation mode of gas-rich galaxies at high redshift (e.g.,
+Gaibler et al. 2012; Silk 2013; Zubovas et al. 2013; Bieri et al. 2015,
+2016), our simulations show that powerful AGN winds acting on a
+massive star-forming galaxy can only trigger a small amount of star
+formation compared to the overall negative effect.
+Some studies propose that the dominant feedback effect, positive
+or negative, may depend on global host galaxy properties and/or
+the physical conditions in different parts of the galaxy. Nayakshin
+(2014) argues that AGN feedback has a negative effect in gas poor
+galaxies but a global positive effect in gas rich galaxies with AGN
+feedback accelerating the collapse of the cold ISM phase, in contrast
+with our results. Gaibler et al. (2012) simulated the impact of AGN
+jets on idealized simulations of high-redshift galaxies, arguing that
+jet feedback suppresses star formation in the central region, forming
+MNRAS 000, 1–16 (2023)
+
+Local positive AGN feedback in the overall negative
+13
+a ring-like star-forming structure at the cavity boundary in qualita-
+tive agreement with our results, but they predict a strong increase
+in global SFR due to the external pressure enhancement throughout
+the galaxy disc produced by the jet cocoon (see also Sutherland &
+Bicknell 2007; Bieri et al. 2015, 2016; Dugan et al. 2017). Using ide-
+alized simulations of the impact of AGN outflows on the fragmen-
+tation rate of gas in turbulent spheres, Zubovas & Bourne (2017)
+proposed that there is a critical AGN luminosity at which positive
+feedback dominates, with outflows being too efficient at removing
+gas at higher luminosities. Different galaxy properties, AGN feed-
+back mechanisms, and efficiencies may thus play a role in the de-
+tailed balance of positive and negative feedback. Our cosmological
+zoom-in simulations including a detailed treatment of star forma-
+tion, stellar feedback, and hyper-refined AGN-driven winds propa-
+gating in a multi-phase ISM suggest that AGN feedback has either
+a minor global impact on massive star-forming galaxies (assuming
+low feedback efficiency) or net negative effects (for high feedback
+efficiency), but the contribution of positive feedback to star forma-
+tion in galaxies under different conditions and/or redshifts should be
+investigated in future work.
+Signatures of local positive AGN feedback in our simulations
+include the spatial anti-correlation of wind-dominated regions and
+star-forming clumps, and higher local star formation efficiency in
+regions compressed by the winds, in qualitative agreement with ob-
+servations (Cresci et al. 2015a,b; Carniani et al. 2016; Shin et al.
+2019; Perna et al. 2020; Bessiere & Ramos Almeida 2022; Schutte
+& Reines 2022). Examples of high redshift galaxies where posi-
+tive and negative AGN feedback may co-exist include SINFONI ob-
+servations of quasars with extended outflows (traced by OIII) that
+appear to suppress star formation in a central cavity while trigger-
+ing star formation (traced by Hα) along the edges of the outflow-
+dominated region (Cresci et al. 2015b; Carniani et al. 2016), qualita-
+tively similar to our simulated galaxy. Examples in the low redshift
+universe include MUSE observations of Seyfert galaxies with simi-
+lar spatial anti-correlations of outflow components and star-forming
+regions (Cresci et al. 2015a; Shin et al. 2019). Combining MUSE
+and ALMA observations, Shin et al. (2019) showed that regions of
+a star-forming ring structure interacting with a large-scale outflow
+in a nearby Seyfert 2 galaxy have higher SFR density but compar-
+atively lower molecular gas content than other regions. This may
+imply higher star formation efficiency by a factor ∼3–5 owing to
+positive AGN feedback, in qualitative agreement with our results.
+Following a different approach, Bessiere & Ramos Almeida (2022)
+compared the spatial distribution of the young stellar populations
+(<100 Myr) of the nearby Type II quasar Markarian 34 and the kine-
+matics of the warm ionized outflows, finding a local enhancement
+of recent star formation coinciding with the outer edge of one side
+of the outflow, while increased turbulence and disrupted gas kine-
+matics with no signs of recent star formation in the other side of the
+outflow. This provides further indication that positive and negative
+AGN feedback can coexist in galaxies, as suggested by our simu-
+lations. Nonetheless, simulated global galaxy SFRs are consistently
+higher in the absence of AGN winds, supporting scenarios where
+AGN feedback is globally negative (if anything) while star forma-
+tion triggering represents a subdominant component.
+In the absence of AGN winds, the star-forming gas reservoir con-
+tains roughly similar fractions of inflowing and outflowing com-
+ponents, reflecting the turbulent ISM dynamics prevalent through-
+out the simulation. However, efficient AGN winds can significantly
+change the ISM gas kinematics. Simulations with AGN winds show
+stronger variability in the contributions of different dynamical com-
+ponents to the global SFR at later stages, varying from up to ∼90% of
+7.5
+8.0
+8.5
+9.0
+9.5
+10.0
+log10( M ) [M ]
+M (t)
+M (t0)
+7.5
+8.0
+8.5
+9.0
+9.5
+log10(M ) [M ]
+Stars from SF gas at t0
+0
+10
+20
+30
+t = t
+t0 [Myr]
+7.5
+8.0
+8.5
+9.0
+9.5
+log10(Mgas) [M ]
+Leftover SF gas from t0
+R < 2 kpc
+No-wind
+m0.1e0.5
+m1e5
+AGN-wind
+m4e20
+m10e50
+Figure 11. Top: Stellar mass growth within the central 2 kpc as a function of
+time for a variety of simulation runs with different AGN feedback strengths.
+Solid lines represent the total mass of stars formed since t0, i.e. excluding any
+pre-existing stars. Middle: The total mass in stars formed by initially star-
+forming gas at t0 (dashed lines). Bottom: Total gas mass remaining from the
+original star-forming gas reservoir (dotted lines). The horizontal dash-dotted
+line (black) represents the initial total mass of star-forming gas available at
+the start of the simulations (t0).
+MNRAS 000, 1–16 (2023)
+
+14
+J. Mercedes-Feliz et al.
+inflowing material to ∼90% of outflowing material over time. Some
+observations suggest that outflows driven by AGN feedback con-
+tain star-forming gas within the outflow itself, which may consti-
+tute a strong manifestation of positive AGN feedback (Santoro et al.
+2016; Maiolino et al. 2017; Cresci & Maiolino 2018; Gallagher et al.
+2019; Rodríguez del Pino et al. 2019). We have identified some ex-
+amples of outflowing star formation material, including a clear out-
+flow component contributing ≳70% of the total SFR (16 M⊙ yr−1 out
+of ∼ 20 M⊙ yr−1) with velocities reaching up to ∼1000 km s−1. How-
+ever, we do not see a consistent trend of outflowing gas dominating
+the SFR; the inflow component actually becomes more prevalent at
+later times, suggesting that gas pushed by the AGN winds has lower
+probability of forming stars except for short transitory phases.
+The small overall amount of star formation in fast outflows sug-
+gests that positive AGN feedback does not contribute much to high
+velocity stars populating the stellar halo, possibly less than similar
+processes operating in galactic winds driven by stellar feedback (Yu
+et al. 2020). This is in contrast with some analytic models where gas
+swept outward by radiation pressure from the AGN efficiently forms
+outflowing stars that drive the size and structural evolution of mas-
+sive galaxies (Ishibashi & Fabian 2012, 2014). Instead, the dominant
+structural effect of efficient AGN winds in our simulations is the sup-
+pression of the central starburst, with the consequent reduction in the
+nuclear stellar density and increase in the stellar effective radius of
+the galaxy (Anglés-Alcázar et al. in prep., Cochrane et al. in prep).
+In a companion paper (Mercedes-Feliz et al., in prep.), we explore in
+detail the role of powerful AGN winds driving the formation of very
+dense stellar clumps in rare but extreme positive feedback events.
+Future work should consider the role of AGN winds on a larger
+sample of galaxies across different halo masses and redshifts, to in-
+vestigate if the results presented here can be generalized beyond the
+specific galaxy conditions simulated here, and to evaluate whether
+host galaxy properties play a role in determining the efficiency
+of positive AGN feedback. In addition, future simulations should
+consider the impact of AGN winds coupled to BH accretion self-
+consistently, unifying our wind particle spawning technique with the
+hyper-refinement scheme presented in Anglés-Alcázar et al. (2021)
+to increase the mass resolution dynamically as gas approaches the
+BH. The BH accretion rate is expected to decrease owing to AGN
+feedback self-regulation (e.g. Di Matteo et al. 2005; Choi et al. 2012;
+Hopkins et al. 2016; Habouzit et al. 2021). Our results, assuming
+constant accretion rate, should thus represent an upper limit to the
+impact of accretion-driven winds for a given kinetic efficiency.
+8 SUMMARY AND CONCLUSIONS
+We have presented a detailed analysis of a set of high-resolution cos-
+mological zoom-in simulations of a massive galaxy near the peak
+of star formation activity (Mhalo ∼ 1012.5 M⊙ at z ∼ 2) to inves-
+tigate the plausible positive versus negative effects of AGN feed-
+back during a luminous quasar phase. Crucially, our simulations
+include resolved multi-phase ISM physics from the FIRE project
+(Hopkins et al. 2018) and a novel implementation of hyper-refined
+AGN-driven winds that simultaneously captures their propagation
+and impact from the inner ≲10 pc to CGM scales (Anglés-Alcázar
+et al., in prep.). Comparing simulations without MBH feedback and
+with different AGN feedback strength, our results can be summa-
+rized as follows:
+• Strong AGN winds with 5% kinetic efficiency, powered by a
+MBH with mass MBH = 109 M⊙ accreting at the Eddington rate,
+drive the formation of a central gas cavity of size ∼200 pc and can
+dramatically reduce the star formation rate surface density across
+the galaxy disc in ∼30 Myr, indicating that powerful quasar winds
+have a global negative impact on star formation.
+• Most of the star formation suppression relative to the control
+simulation without AGN winds is driven by a strong reduction
+in the amount of new gas that can become star-forming during a
+period of intense gas accretion onto the galaxy. Direct ejection
+of the pre-existing star-forming gas reservoir at the beginning of
+the quasar phase only accounts for a small fraction of the overall
+reduction in stellar growth, suggesting that preventive feedback
+dominates over ejective feedback for strong AGN winds.
+• Reducing the kinetic efficiency by a factor of ten (ϵk = 0.5%)
+barely affects the star-forming rate of the host galaxy, with no clear
+signs of either global positive or negative feedback, while increasing
+the kinetic efficiency by a factor of ten (ϵk = 50%) results in a
+complete blow out of the star-forming gas reservoir in ≲10 Myr.
+• Compression of gas near the edge of the cavity by AGN winds can
+increase the local star formation rate surface density compared to
+the same radial range in the simulated galaxy without AGN winds.
+This local triggering of star formation is driven by the increased
+amount of gas accumulated at the edge of the cavity as well as the
+higher star formation efficiency of compressed gas.
+• The spatial anti-correlation of wind-dominated regions and star-
+forming clumps along with higher local star formation efficiencies
+constitute plausible signatures of local positive AGN feedback, in
+qualitative agreement with observations. However, the highest star
+formation efficiency reached across all simulations occurs in the
+absence of AGN winds, owing to the much larger central stellar
+mass surface density and the inability of stellar feedback to regulate
+star formation.
+• Besides the overall suppression of star formation, efficient AGN
+feedback drives a spatial and temporal shift in star formation. Stars
+that do form under the presence of AGN winds tend to do so at
+larger radial distances compared to stars formed out of the same gas
+clumps in the absence of AGN winds. Star formation also tends to
+occur earlier in time at the beginning of the quasar phase, with AGN
+winds triggering faster conversion of gas into stars in local regions,
+but comparatively later in time for stars forming after ∼20 Myr of
+global negative impact of AGN winds.
+• The main impact of AGN feedback on the kinematics of new
+stars is the reduction of their typical radial velocity (either positive
+or negative). The strong deepening of the nuclear gravitational
+potential in the absence of AGN winds results in stellar radial
+velocities reaching up to 1000 km s−1, while the suppression of the
+nuclear stellar density by AGN winds yields stellar radial velocities
+≲ 500 km s−1.
+• AGN winds tend to produce decoupling of dynamical components,
+with star-forming gas dominated by either inflowing or outflowing
+material in different regions. In some cases, the local contribution of
+outflowing material to star formation can exceed that of the inflow
+component by factors >10. However, the inflow component tends
+to dominate at later times, suggesting that gas radially accelerated
+by the winds has lower probability of forming stars except for short
+transitory phases.
+MNRAS 000, 1–16 (2023)
+
+Local positive AGN feedback in the overall negative
+15
+In conclusion, our results suggest that positive and negative AGN
+feedback coexist in galaxies, where strong quasar winds can sup-
+press global stellar mass growth while locally triggering a small
+amount of star formation. The simulations presented here do not sup-
+port scenarios where positive AGN feedback is the dominant mech-
+anism powering rapid star formation in galaxies.
+ACKNOWLEDGEMENTS
+The simulations were run on Flatiron Institute’s research comput-
+ing facilities (Gordon-Simons, Popeye, and Iron compute clusters),
+supported by the Simons Foundation. We thank the Scientific Com-
+puting Core group at the Flatiron Institute for outstanding support.
+Additional numerical calculations were run on the Caltech compute
+cluster “Wheeler,” allocations FTA-Hopkins supported by the NSF
+and TACC, and NASA HEC SMD-16-7592, and XSEDE allocation
+TG-AST160048 supported by NSF grant ACI-1053575. JMF was
+supported in part by a NASA CT Space Grant Graduate Fellow-
+ship. DAA acknowledges support by NSF grants AST-2009687 and
+AST-2108944, CXO grant TM2-23006X, and Simons Foundation
+award CCA-1018464. SW was supported by an NSF Astronomy and
+Astrophysics Postdoctoral Fellowship under award AST2001905.
+CAFG was supported by NSF through grants AST-1715216, AST-
+2108230, and CAREER award AST-1652522; by NASA through
+grants 17-ATP17-006 7 and 21-ATP21-0036; by STScI through
+grants HST-AR-16124.001-A and HST-GO-16730.016-A; by CXO
+through grant TM2-23005X; and by the Research Corporation for
+Science Advancement through a Cottrell Scholar Award. KS ac-
+knowledges support from the Black Hole Initiative at Harvard Uni-
+versity, which is funded by grants from the John Templeton Founda-
+tion and the Gordon and Betty Moore Foundation, and support from
+Simons Foundation.
+DATA AVAILABILITY
+The data supporting the plots within this article are available
+on reasonable request to the corresponding author. FIRE-2 sim-
+ulations are publicly available (Wetzel et al. 2022) at http://
+flathub.flatironinstitute.org/fire. Additional FIRE sim-
+ulation data, including initial conditions and derived data prod-
+ucts, are available at https://fire.northwestern.edu/data/.
+A public version of the GIZMO code is available at http://www.
+tapir.caltech.edu/~phopkins/Site/GIZMO.html.
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+APPENDIX A: STAR FORMATION EFFICIENCY PER
+FREE-FALL TIME
+Figure A1 shows the radial profile of the star formation efficiency
+(SFE), as in Figure 6, and the SFE per free-fall time (ϵff) for two
+snapshots, ∆t = 10 Myr and ∆t = 20 Myr since the start of the quasar
+feedback phase. We compute ϵff ≡ (SFR/Mgas) × tff = SFE × tff
+for each gas element according to its mass Mgas and SFR, where
+the free-fall time is defined as tff =
+�
+3π/32Gρ. We then compute
+the mass-weighted average over all gas in cylindrical radial bins.
+The grey bands correspond to regions where ΣSFR is higher in the
+AGN-wind simulation compared to the no-wind case, which coin-
+cides with higher SFE and higher molecular gas mass surface den-
+sity in the presence of AGN winds (Figure 6). Here, we see that
+these regions also have higher effective star formation efficiency per
+free-fall time ϵff, recovering similar results. In the FIRE simulations,
+ϵff is a predicted quantity (Hopkins et al. 2018), with typical kpc
+scale-averaged efficiencies in the range ϵff ∼ 0.01 − 0.1 for Milky
+Way-mass galaxies at z = 0 (Orr et al. 2018), determined by stel-
+lar feedback self-regulation (Ostriker & Shetty 2011; Hopkins et al.
+2012; Faucher-Giguère et al. 2013). Our simulations of a massive,
+high-redshift galaxy represent very different conditions either with
+our without AGN winds. In the no-wind simulation, we find that ϵff is
+significantly higher in the nuclear region because stellar feedback is
+no longer able to regulate star formation with such high stellar sur-
+face density; as expected, the average ϵff on kpc scales is much more
+similar to low-z Milky Way-mass galaxies. In the AGN-wind case,
+the ultra-compact nuclear region does not develop, and the efficiency
+remains in the range ϵff ∼ 0.01 − 0.1.
+This paper has been typeset from a TEX/LATEX file prepared by the author.
+MNRAS 000, 1–16 (2023)
+
+Local positive AGN feedback in the overall negative
+17
+3.0
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+log10(R) [kpc]
+10
+9
+8
+7
+6
+5
+log10(SFE) [yr
+1]
+6
+5
+4
+3
+2
+1
+0
+log10( ff)
+6
+5
+4
+3
+2
+1
+0
+log10( ff)
+t = 10.0 Myr
+No-wind
+AGN-wind
+SFE
+ff
+3.0
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+log10(R) [kpc]
+10
+9
+8
+7
+6
+5
+log10(SFE) [yr
+1]
+6
+5
+4
+3
+2
+1
+0
+log10( ff)
+6
+5
+4
+3
+2
+1
+0
+log10( ff)
+t = 20.0 Myr
+No-wind
+AGN-wind
+SFE
+ff
+Figure A1. Radial profile of the star formation efficiency (SFE; solid lines) and the SFE per free-fall time (ϵff; dashed lines) at time ∆t = 10 Myr (left) and
+∆t = 20 Myr (right) since the beginning of the quasar feedback phase for the no-wind (blue) and AGN-wind (orange) simulations. The vertical grey band
+indicates the main region where SFE and ϵff are higher in the AGN-wind simulation compared to the no-wind case, which also corresponds to a region with
+higher ΣSFR in the presence of AGN winds.
+MNRAS 000, 1–16 (2023)
+

UtAyT4oBgHgl3EQfhfhI/content/tmp_files/2301.00377v1.pdf.txt

@@ -0,0 +1,748 @@
+ITHACA. A TOOL FOR INTEGRATING FUZZY
+LOGIC IN UNITY
+1st Alfonso Tejedor Moreno
+Department of Computer Science
+University of Almeria
+Almeria 04120, Spain
+alfi[email protected]
+2nd Jose A. Piedra-Fernandez
+Department of Computer Science
+University of Almeria
+Almeria 04120, Spain
+[email protected]
+3rd Juan Jesus Ojeda-Castelo
+Department of Computer Science
+University of Almeria
+Almeria 04120, Spain
+[email protected]
+4th Luis Iribarne Member, IEEE
+Department of Computer Science
+University of Almeria
+Almeria 04120, Spain
+[email protected]
+Abstract—Ithaca is a Fuzzy Logic (FL) plugin for developing
+artificial intelligence systems within the Unity game engine. Its
+goal is to provide an intuitive and natural way to build advanced
+artificial intelligence systems, making the implementation of such
+a system faster and more affordable. The software is made up
+by a C# framework and an Application Programming Interface
+(API) for writing inference systems, as well as a set of tools
+for graphic development and debugging. Additionally, a Fuzzy
+Control Language (FCL) parser is provided in order to import
+systems previously defined using this standard.
+Index Terms—Artificial Intelligence, Fuzzy Logic, Fuzzy Con-
+trol Language, video games.
+I. INTRODUCTION
+G
+RAPHICS is a relevant factor in terms of video games
+and that fact is appreciated in the latest development of
+video games and consoles. Each new generation of consoles
+empower their resources such as a graphic card with a lot
+of calculation, a processor with many cores and so forth.
+Therefore, the video games are developed with very realistic
+graphics and the users feel that they are part of a real
+environment more than a virtual world. However, users are
+more demanding and currently artificial intelligence (AI) plays
+a significance role in this field since a video game is not really
+popular if its graphics are spectacular but its AI is very low.
+In fact, the quality of AI is an important feature in order to
+increase the immersion and enjoyment of the players [1].
+The AI has been a magnificent contribution to the video
+games sector since it has mastered games as Chess [2] or Go
+[3]. Or even the algorithm developed by DeepMind which
+is able to cooperate with human players or other machine
+players [4]. These agents have been tested in the game Quake
+III. According to Pirovano [5], the two most common AI
+techniques found in games are Finite State Machines (FSM)
+and Decision Trees (DT), far from the performance of more
+complex techniques like Artificial Neural Networks (ANN)
+and Genetic Algorithms (GA). Therefore our main goal is
+making AI achievable for small and medium sized teams, pro-
+viding a tool for implementing a technique halfway between
+the easy to use of the FSM and the power and flexibility of
+the ANN. And right from the most used game engine, Unity 1
+(Fast Facts - Unity) The AI in games is not only for the NPC
+or that the player can make more decisions in the game if not
+that AI is being used to remaster video games and improve
+the quality of these products in hours.
+On the other hand, most game developers work on huge
+constraints of time and budget, thus the AI implemented
+specifically in games by indie developers is naive at best, if
+there is any at all. That is the reason that we have developed a
+plugin with fuzzy logic in order to help the developers and so
+forth to apply fuzzy logic in their projects in a easy way and
+make the most of it. Moreover, this plugin was developed for
+Unity because this game engine is operated from students or
+hobbyists to professionals and game studios. Unity allows the
+developers to create video games in multiple platforms, more
+exactly 30 different platforms, such as mobile, PC or console.
+As a result, this tool is very useful for people who wants
+to learn how to develop games, people that they only want to
+create a video game as a hobby or even in a more professional
+level entrepreneurs who wants to set up their entertainment
+company or create a professional game. Furthermore, this
+plugin has been designed with the fundamental functions of
+fuzzy logic and is very easy to use because it is aimed at
+beginners without previous knowledge of the subject. And
+this is possible because Unity is adapted to many needs being
+its learning process very easy at the beginning although it
+gets more difficult if you want to do a complex game. These
+features make that this game engine is used by a wider
+population than other game engines as Unreal Engine 2 In
+fact, John Riccitiello (Unity CEO) states that the half of all
+the games has been developed on Unity 3.
+The objectives of the project are:
+1Unity - https://unity.com/
+2Unreal Engine - https://www.unrealengine.com/en-US/
+3Unity games developed - https://techcrunch.com/2018/09/05/unity-ceo-
+says-half-of-all-games-are-built-on-unity/
+arXiv:2301.00377v1  [cs.AI]  1 Jan 2023
+
+This is a preprint and it is not the ultimate version.
+• Create an easy to use tool that makes the Artificial Intelli-
+gence and Fuzzy Logic available to all game developers.
+• The tool has to be simple and flexible in order to be used
+in any kind of projects and games.
+• The user must be able to work with the tool using either
+a Graphical User Interface (GUI) or an API.
+• Be compliant with the FCL standard as defined in IEC
+61131-7 [6]
+The remainder of the paper is organized as follows: Section
+II presents some background on Fuzzy Logic and Artificial In-
+telligence. Section III describes the architecture and its Fuzzy
+Logic foundations. Section IV shows the results obtained
+by building a few small games. Section V summarizes the
+conclusions and discusses the future work.
+II. BACKGROUND
+In 1940 Edward Uhler Condon presented a Math game at
+the exhibition of Westinghouse. This human vs machine game
+was played by thousands of people of which only 10% were
+able to beat the machine. However, it was not until the 1970s
+that video games began to commercialize on a large scale, as
+in the case of the game developed by the Atari company: Pong
+[7]. At that time, Neural Networks were not included in the AI
+system of the video game, although simpler techniques such as
+finite state machines were used. The next section will briefly
+describe the influence of AI in the world of video games.
+A. Artificial Intelligence in Games
+In 1959 Arthur Lee Samuel introduced Artificial Intel-
+ligence into the world of video games, more specifically
+teaching a computer how to play the game of checkers [8].
+Thereafter, other games were developed that included artificial
+intelligence, highlighting the video game Space Invaders in the
+70s and Pac-Man in the 80s. In Space Invaders, complexity
+was added to the game by introducing hash functions that
+”learned” from the player’s actions. The heart of the Pac-
+Man game is the behavior of the ghosts that are the player’s
+enemies, where each one has a different behavior [9] and it is
+controlled by a state machine [10]. In the 90s video games
+were booming due to the appearance of various consoles
+such as Gameboy, SNES or PS One. This expansion also
+produced the inclusion of more complex AI systems, with
+special mention to the Half-life video game in which it existed
+a cooperation between the enemies to kill the player [11].
+In the 21st century AI has advanced considerably and its
+algorithms are capable of defeating a human player in games
+like Starcraft II [12], [13].
+All these developments have in common, regardless of the
+AI that they integrated, the improvement of the user experience
+and for this objective AI is an outstanding candidate. AI can
+be used in different aspects of a video game as pathfinding,
+movement, making decision or learning [10], but probably
+the one that stands out the most is the one that controls
+the behavior of NPCs. The reason is because if the behavior
+of these elements is realistic and resembles the real world,
+the users will feel that he is not playing in an artificial and
+predefined world but in a world more similar to their own. In
+reference to this last idea, it is important to remember that the
+aim of the AI is not to beat the player always, since the fun and
+motivation would be lost, but rather to create an experience as
+realistic as possible.
+There are even studies that have analyzed the feasibility of
+Deep reinforcement learning for serious games [14]
+In addition to Deep reinforcement learning, genetic algo-
+rithms have brought improvements in this area. In [15] AI
+characters for fighting games can be generated with the help of
+genetic algorithms and through a low-cost process. Moreover,
+it has also been used in a soccer simulator (Robocup Soccer)
+with the aim of improving decision making in the simulation
+of soccer teams [16]. The result of the implementation of this
+enhancement was that two of the three teams that made up
+this system were top teams in the RoboCup competition. In
+[17] sets out to apply genetic algorithms in real-time strategy
+games to learn new and effective strategies for this type of
+game. The problem is that these types of actions are usually
+abstract for the player and for this reason in this work they
+focus on learning strategies that can be easily interpreted by
+people. Furthermore, this type of algorithms have also been
+used in games such as Backgammon [18] or Chess [19].
+In the previous paragraphs, a series of applications have
+been shown that are far from the AI techniques used in the
+first video games, which were defined by rule-based systems,
+that constitute the most basic algorithms in AI. More complex
+and innovative structures such as Deep learning or genetic
+algorithms are currently integrated. However, there is an AI
+technique called fuzzy logic that, despite not being as popular
+as neural networks, is very useful for this type of development
+and will be described below.
+B. Fuzzy Logic
+Fuzzy Logic is advisable for solving many problems where
+the uncertainty is a main actor, for instance the uncertain
+multi-criteria decision making problem [20]. It is broadly
+used in controllers where it is hard to obtain a mathematical
+model, or systems that need to work with some vagueness or
+ambiguousness. Some applications created with fuzzy logic
+would be medical applications [21], daily life applications
+as the washing machine [22], natural interaction with face
+recognition [23] and posture recognition [24] and lastly smart
+cities. Perhaps the last one could be disconcerting, notwith-
+standing the growth of cities has generated various problems in
+them. The administration of aspects such as waste collection,
+environmental pollution, traffic, energy distribution or water,
+becomes an increasingly complex task. Fuzzy logic has con-
+tributed to energy efficiency in smart cities [25].
+1) Fuzzy Logic in Games: Fuzzy logic, like other AI
+techniques cited previously, has also contributed to video
+games. In special needs, games have been developed specially
+for autistic people. In [26] FL is used in order to rate the
+social skills in autistic children and adapt the level of the
+game to them. In the diagnoses of a patient’s autism level,
+the patient has to carry out activities that will determine that
+2
+
+This is a preprint and it is not the ultimate version.
+level. The authors of this work [27] have designed a game
+that is able to calculate the player’s autism level using FL.
+Nevertheless, apart from carrying out work with people with
+autism, serious games have also been designed for people with
+physical impairments, such as the example of this game called
+ReHabGame [28] where a series of activities are conducted
+for the upper and lower limbs, in order to evaluate and
+improve motor and sensory faculties in patients who have
+neuromuscular problems. This serious game allows the user
+to interact with Kinect in order to obtain information on the
+movement of their body during the performance of tasks and
+thus the fuzzy logic system will be able to extract and analyze
+the data corresponding to the positions of the users to satisfy
+their needs. In addition to ReHabGame, other serious games
+with different themes have also been developed, such as this
+serious game, in which the authors have studied the variables
+of throttle position sensor, engine rotation speed and car speed
+to determine by FL whether the driving style is efficient and in
+this way improve driver behavior in front of the steering wheel
+[29]. The authors expanded the work done integrating FL and
+Random Forest in order to know which algorithm was best
+for this type of serious game. The results indicated that the
+advantage of FL was that it produced understandable linguistic
+feedback while Random Forest predicted fuel consumption
+more accurately [30]. Another work that is worth mentioning
+for its novelty is the study conducted in [31] which aims to
+create a game to induce emotions in students with the aim
+of improving their learning process through inductive control.
+This system integrates fuzzy logic to analyze the performance
+and emotional state of the players through a voice analysis.
+C. AI Plugins for Unity
+Recently the Unity engine has finally embraced artificial
+intelligence, adding Machine Learning (ML) to its pathfinding
+tool. The most outstanding open-source plugin that this game
+engine has it is called ML-Agents. This plugin enables games
+and simulators to become a training environment to train
+artificial intelligence agents through reinforcement learning
+[32]. Anyhow, if a developer wants to implement another
+technique the first stop is the Unity’s Assets Store, filled
+with thousand of plugins. But among all the variety of tools
+available, those intended for implementing AI are scarce, and
+only one of them is based on fuzzy logic. This plugin is just
+a Fuzzy Logic layer for another tool aimed to build fighting
+games. The best solution is using an external library called
+AForge.net (AForge.NET). Written in C#, is an AI library
+that includes an API for working with Neural Networks,
+Artificial Vision, and Machine Learning, among others. Given
+the broad scope of the library, the Fuzzy Logic implementation
+is shallow but easy to use. It has no Unity integration, so
+it lacks of GUI and makes it harder to use by staff like
+designers. The project is open source with a LGPLv3 license,
+so it can be easily extended, but our tool already has most
+of the elements it lacks like more membership functions,
+operators, or defuzzification methods. Ithaca was built from
+scratch thinking in Unity, so it implements components that
+can be attached to game objects. It also has a GUI for defining
+a system and debugging it without writing a single line of
+code. All these makes our solution more complete, more
+powerful and flexible, and thanks to its integration with Unity,
+easier to use.
+III. ITHACA
+A. What is Ithaca
+Ithaca is a tool for integrating a Fuzzy Logic Inference
+System (FIS) on any project developed with the Unity engine.
+It is written in C# but it is not compiled, so all code is available
+to the developer for any needed modification. This makes the
+tool platform-independent, allowing to use it in any device
+among the ones available in Unity. Its core consist in a Fuzzy
+Rule Based System (FRBS) that evaluates a set of rules which
+are formed by fuzzy sets and fuzzy logic.
+B. Fuzzy Logic and Inference Engine
+The fuzzy inference system implemented in Ithaca uses the
+Mamdani model, where the consequent of the IF-THEN rules
+is a fuzzy statement like this:
+IF X is A and Y is B THEN Z is C
+This way of writing rules is natural and intuitive, making it
+more suitable for developing systems where we are trying to
+simulate human decision making. This model is less efficient
+than the Sugeno model, where the consequent of the rules are
+algebraic expressions [33] so the defuzzifying step is faster to
+compute. For defuzzifying Mamdani FIS we implemented the
+next methods:
+• Centroid or Centre of Gravity (COG):
+COG =
+� max
+min xµ(x)dx
+� max
+min µ(x)dx
+• Centroid for Singleton (COGS):
+COGS =
+�
+i xiµi
+�
+i µi
+• Bisector or Centre of Area (COA):
+COA = u′,
+� u′
+min
+µ(u)du =
+� max′
+u′
+µ(u)du
+• Mean of Maximum (MOM):
+MOM =
+�
+Amax xdx
+�
+Amax dx
+• Right Most (RM):
+RM = {x|x > x′, ∀x′ ∈ Amax}
+• Left Most (LM):
+LM = {x|x < x′, ∀x′ ∈ Amax}
+For the fuzzifying interface we wrote twelve membership
+functions, being some of them specific cases from more
+general functions, but broadly used. Regarding the logical
+3
+
+This is a preprint and it is not the ultimate version.
+connectives for the rules’ elements, conjunction can be defined
+by a set of different t-norm methods. The same happens to the
+union, which may be defined using one of various t-conorms.
+All the methods implemented are:
+• Membership functions:
+– Singleton.
+– Piecewise.
+– Triangle.
+– Trapezoid.
+– Grade.
+– Reverse grade.
+– Gaussian.
+– Double Gaussian.
+– Bell.
+– Cosine.
+– Sigmoidal.
+– Difference of sigmoidals.
+• T-Norm (AND operation):
+– Min (G¨odel-Dummett).
+– Product
+– Bounded Difference (Łukasiewicz).
+• T-CoNorm (OR operation):
+– Max.
+– ASUM.
+– BSUM.
+– NSUM.
+C. Architecture
+The system architecture is divided in three subsystem;
+the core, the graphics subsystem, and the parser. The core
+subsystem has three main components (Figure 1):
+Fig. 1: System architecture
+1) The fuzzifier interface: this is where the crisp input
+values are turned into fuzzy values depending on the
+membership functions.
+2) The inference system: all the rules are evaluated here.
+The antecedents determine the value of the consequences
+and the result is a fuzzy value.
+3) The defuzzifier interface: the outputs are turned into
+crisp values in order to be used in the game.
+The graphics subsystem is formed by the classes for the GUI
+and an additional set of classes, in a separated namespace, for
+drawing the graphs for the membership functions (Figure 3).
+Unity has no built-in system for drawing graphs, so we had to
+build our own plugin. We made it decoupled from the rest of
+the GUI so it could be used in other cases. At last, the parser
+subsystem translates the FCL files to the Ithaca inner format.
+The inference engine structure is based in the FCL’s def-
+inition, where a system is represented by a Function Block
+(FB), which in turn is defined by a set of Linguistic Variables
+(LV) as inputs and outputs, and a set of rules or Rule Block
+(RB). Each LV represents a domain of knowledge, and if
+declared as input has as many fuzzy sets as needed. Each
+fuzzy set has a membership function that defines the degree of
+correspondence of an input value to this set. On the other side,
+if a LV is declared as output, it has an additional defuzzifier
+method to translate the fuzzy values to crisp values. The rules
+set is changeable in runtime, making it possible to adapt the
+knowledge base depending of the needs.
+Once a system is defined using the API, GUI, or a FCL file,
+the inference engine can be called during gameplay asking for
+an output depending on the current value, or with a new value.
+The GUI has a main window with three tabs (see Figure 2):
+• One for creating/loading a system
+• Another one for defining input and output variables
+• The last one to write sets of rules
+Fig. 2: Ithaca window
+The first tab lets the user create a new system or import
+one already defined. The systems can be exported to JSON
+format, and be loaded back from JSON and also from an
+FCL file. This lets the users save and exchange systems in a
+format trackable by version control software. Once the system
+is created, an asset file is created in the Unity project. This is
+a binary file with a serialization of the full system. If the asset
+file is selected, it can be edited from this window. The second
+tab is for declaring the input and output linguistic variables.
+Once the LV is created, it can be edited in order to add the
+fuzzy sets and its membership functions (Figure 3). The user
+can declare the limits of the LV and its default value.
+Finally, in the third tab, the user can define sets of rules
+written in natural language (Figure 4). For each set of rules the
+user can decide the methods for the operations. This operations
+are:
+4
+
+Ithaca Fuzzy Logic Editor
+System info
+Variables
+Rules
+Info
+Import/Export
+Name:container_crane
+Save as JSON
+LoadfromJSON
+Load from FCLThis is a preprint and it is not the ultimate version.
+Fig. 3: Linguistic variable editor
+• The methods for the logic operations AND/OR, also
+called aggregation operation. In order to satisfy De Mor-
+gan’s law the methods must go in the following pairs:
+– Min-Max
+– Prod-ASUM
+– BDIF-BSUM
+• The methods for the activation operation. This operation
+translates the result of the antecedent to the consequent.
+– Min
+– Prod
+• The method for the accumulation operation. This oper-
+ation takes the results from the consequent of all fired
+rules and combines its values to obtain one output.
+– Max
+– BSUM
+– NSUM
+If the used defines more than one set of rules, a default one
+must be choosen. As we said, the RB can be changed during
+runtime.
+Fig. 4: Rules editor
+There is also an additional window, the debugger. It lets the
+developer debug both the graphs values and the rules that fired
+and its values. (Figure 5).
+Fig. 5: Debug window
+D. Features
+The main features of the Ithaca tool are:
+• A powerful GUI that allows to define a full system using
+graphs.
+• The inference system used is Mamdani, which uses fuzzy
+sets as outputs for the rules. This way the output is easier
+to use by the developer in most cases.
+• Covers the most used membership functions, defuzzifier
+methods, and operation expressions. The user can define
+new functions, methods, or expressions.
+• It is not compiled, thus the user can modify the code as
+needed within Unity.
+• A graphical debugging system that allow to check in real
+time the input and output values, as well as the values of
+every rule and if it was fired.
+• It can be used to build games in any platform available
+in Unity.
+IV. RESULTS
+A. Crane
+The goal of this test was to build a complex system using
+our tool, and to check the compliance with the FCL standard.
+So we decided to use the crane system defined in the FCL
+standard [6], that controls container crane that loads and
+unloads containers from ships. The controller must moves the
+containers taking care of the speed and angle(Figure 6). The
+rules of the system were:
+1) IF distance IS far AND angle IS zero THEN power IS
+pos medium.
+2) IF distance IS far AND angle IS neg small THEN power
+IS pos high.
+5
+
+0.75
+0.5 
+0.25
+1.25
+2.5
+3.75
+门
+Rocket_Ammunition Sets
+Low:Inverse Grade function
+Ok: Triangular function
+High: Grade functionIthaca Fuzzy Logic Editor
+Systeminfo
+Variables
+Rules
+Default:
+Nol
+AND/OR:
+MIN- MAX
+ACT:
+MIN
+ACCUM:
+MAX
+Nol rule block
+O:IF distance Is farANDangle IS zero THEN power IS pos_medium
+1:IF distanceIS farANDangle IS neg_small THENpowerISpos_high
+2:IF distance IS far AND angle IS neg_big THEN power IS pos_medium
+3:IF distance IS medium AND angle IS neg_small THEN powerIS neg_medium
+4: IF distance IS closeAND angle IS pos_small THENpowerISpos_medium
+5: IF distance IS zero AND angle IS zero THEN power IS zero
+NewRuleblock
+Delete Ruleblock
+Ithacav.1.ob.AlfonsoTejedorMoreno2ol6Ithaca Debugger
+Graphs
+Rules
+州
+Distance (10)
+1
+0.75
+0.5
+0.25
+2.5
+7.5
+10
+Ammunition (10)
+0.75
+0.5
+0.25
+10
+15
+20
+Desirability (0.1941917)
+0.75
+0.5
+0.25
+0.25
+0.5 
+0.75
+1This is a preprint and it is not the ultimate version.
+3) IF distance IS far AND angle IS neg big THEN power
+IS pos medium.
+4) IF distance IS medium AND angle IS neg small THEN
+power IS neg medium.
+5) IF distance IS close AND angle IS pos small THEN
+power IS pos medium.
+6) IF distance IS zero AND angle IS zero THEN power IS
+zero.
+The test demonstrated both the compliance with the FCL
+standard and the usefulness of the tool for building complex
+system using natural knowledge provided by specialists like a
+crane operator.
+Fig. 6: Crane simulation
+B. Race car
+The objective in this test was to develop an adversary AI
+that looks natural and realistic, and to develop it entirely with
+the GUI. Many times it is important to keep the illusion that
+we are playing against a human to engage with the player,
+even if it makes our AI less optimal. The vagueness of the
+Fuzzy Logic makes it ideal for developing systems that act as
+imprecise as a human would. In this case we built two race
+cars, both with the same sensors and almost the same set of
+rules. But one, called ”Classic car”, used classic logic, while
+the ”Fuzzy car” used our inference system. The knowledge
+base used was:
+1) IF Front IS Normal THEN Steering IS Straight, Speed
+IS Somewhat Fast.
+2) IF Front IS Far THEN Steering IS Straight, Speed IS
+Very Fast.
+3) IF Left IS Close THEN Steering IS Right.
+4) IF Right IS Close THEN Steering IS Left.
+5) IF VeryLeft IS Close THEN Steering IS VeryRight.
+6) IF VeryRight IS Close THEN Steering IS VeryLeft.
+The results were that the fuzzy car was faster and its
+trajectory was much more natural (Figure 7). As for the use
+of the GUI, the system revealed itself as faster than using the
+API.
+Fig. 7: Paths followed by AI cars
+V. CONCLUSIONS
+After the experience of developing those tests and some
+more small games, the power of our tool was clear. Its
+installation and integration on the project are really simple
+and straightforward, making the first step as easy as possible
+for any developer, especially the ones less experienced. This
+point is noteworthy as many of the users of Unity are not
+seasoned programmers.
+The use of the GUI or the FCL makes building a Fuzzy
+Logic AI more approachable for small teams or solo develop-
+ers, who can spend more time designing the behaviour of the
+agents rather than writing code. The use of natural language
+for writing the rules and defining the variable makes it easier
+to involve staff that does not know how to code but has a great
+knowledge of the matter. This allows these teams to focus on
+what is perceived by the players, the gameplay and the way
+of acting of its artificial intelligence.
+Not only is it easy to develop, but it is also very simple
+to debug and maintain. Using natural language the developer
+is able to understand what is happening in the system at any
+moment and know what to expect from it. The debugging
+window is additional help on this task, as the user can read
+the status of the system and its expected outcomes in a visual
+way.
+A. Future Work
+• To implement Combs method for reducing the number of
+rules needed by combining them.
+• To add the option to create Fuzzy Finite State Machines
+(FuFSM). The FSM is a very popular technique among
+game developers and the use of FL can provide a degree
+of uncertainty very valuable in some cases.
+• To generate the knowledge base implementing Adaptive
+Network-based Fuzzy Inference System (ANFIS) [34].
+This would allow creating more powerful systems thanks
+to the use of NN.
+6
+
+Distance:12.03
+Angle:-32.12
+Motor power: 2.48Fuzzy car
+Classic carThis is a preprint and it is not the ultimate version.
+• To implement Takagi-Sugeno method for inference.
+• To implement the new Fuzzy Markup Language (FML),
+based on XML [35].
+ACKNOWLEDGMENT
+This work was funded by the EU ERDF and the Spanish Min-
+istry of Economy and Competitiveness (MINECO) under the
+Project TIN2017-83964-R. This work also received funding
+from the CEiA3 and CEIMAR consortium.
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+7
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+Deep Planar Parallax for Monocular Depth Estimation
+Haoqian Liang1
+Zhichao Li2
+Ya Yang1*
+Naiyan Wang2
+1Beijing University of Posts and Telecommunications
+2TuSimple
+{lianghq, yangya}@bupt.edu.cn, {leeisabug, winsty}@gmail.com
+Abstract
+Depth estimation is a fundamental problem in the per-
+ception system of autonomous driving scenes. Although au-
+tonomous driving is challenging, much prior knowledge can
+still be utilized, by which the sophistication of the problem
+can be effectively restricted. Some previous works intro-
+duce the road plane prior to the depth estimation problem
+according to the Planar Parallax Geometry. However, we
+find that their usages are not effective, leaving the network
+cannot learn the geometric information. To this end, we an-
+alyze this problem in detail and reveal that explicit warping
+of consecutive frames and flow pre-training can effectively
+bring the geometric prior into learning. Furthermore, we
+propose Planar Position Embedding to deal with the intrin-
+sic weakness of plane parallax geometry. Comprehensive
+experimental results on autonomous driving datasets like
+KITTI and Waymo Open Dataset (WOD) demonstrate that
+our Planar Parallax Network(PPNet) dramatically outper-
+forms existing learning-based methods.
+1. Introduction
+Depth perception is at the core of 3D computer vision
+task. Benefiting the data-driven approach, recent work has
+shown that learning-based methods [1,4,5,9,21,24,31,55]
+can estimate the geometric properties such as depth or pose
+in one single forward pass. However, pure deep learning
+based methods usually suffer from serious generalization
+problems [25,50,57,65], and not as robust as traditional the-
+oretically sound approaches. Consequently, recent methods
+introduce geometric constraints or priors into deep learning
+approaches to regularize the solution space. For example,
+[61, 63] utilize epipolar geometry for better generalization
+and scale consistency in depth estimation. [26,30] incorpo-
+rate Singular Value Decomposition (SVD) into deep point
+cloud registration to get the final pose.
+For the depth estimation problem in the autonomous
+driving system, the following crucial assumptions exist: the
+*Corresponding author.
+Figure 1. The first row shows current frame superimposed by the
+homography warping of previous frame based on planar paral-
+lax geometry, in which if the pixel is higher than the plane, the
+shearing is heavier. The second row shows the flow estimated by
+a pre-trained optical flow model between the two images in first
+row. Furthermore, the last row displays our depth estimation re-
+sults based on the estimated flow.
+objects and obstacles are all placed on a known road plane,
+and we could access multiple consecutive frames and know
+ego-motion between the frames. This assumption naturally
+leads to the planar parallax (P+P) geometry [40,43]. In par-
+ticular, The P+P geometry shows that the 3D scene struc-
+ture after the alignment of consecutive frames to the refer-
+ence plane can be derived from the residual pixel displace-
+ments caused by the camera’s motion. As the formulation
+is shown in Sec. 3, P+P methods save the capacity for esti-
+mating the depth of ground and eliminate the dependencies
+in estimating camera rotation. Consequently, [53, 59] fol-
+lows the P+P framework, using a neural network to predict
+pixel-wise γ, which is the ratio of height to depth rather than
+a depth map. Nevertheless, we argue that they are reason-
+able attempts, but it does not enjoy the gains of geometric
+1
+arXiv:2301.03178v1  [cs.CV]  9 Jan 2023
+
+priors. It is unachievable to guide the model to learn geo-
+metric information only depending on the supervision of γ.
+We perform detailed experiments to investigate whether the
+P+P pipeline benefits the depth estimation network.
+In this paper, we are committed to fully utilising the P+P
+geometric priors. Intuitively, additional parallax supervi-
+sion can force the network to learn the residual displace-
+ment, which can derive the γ. However, parallax ground
+truth in real-world data is costly, so we choose a more easy
+but practical way. We reveal that a well pre-trained optical
+flow network can keep the ability to estimate the residual
+pixel displacements, dramatically improving depth estima-
+tion results. Regarding the challenge of moving objects, be-
+cause it violates the rigid motion assumption of P+P, we fol-
+low DfM [51] to involve a single view path with learnable
+weights to compensate. Moreover, as derived about γ and
+depth, the depth estimation will be very sensitive to pixel
+error for pixels higher than the camera or near the epipole.
+To remedy this issue, we propose Planar Position Embed-
+ding (PPE), which makes the network aware of the relative
+position to the reference plane of each pixel, guiding the
+network to choose whether to depend more on geometry or
+statistics.
+To summarize, our main contributions are three-folds:
+• We propose a deep planar parallax depth estimation
+network, which effectively incorporates planar paral-
+lax geometry pipeline and data-driven methods. For
+the first time, we demonstrate that optical flow pre-
+training is crucial for using geometric prior.
+• We also propose a Planar Position Embedding which
+introduces the pixel position related to the reference
+plane into the network. It overcomes the inherent de-
+fect of the learning based planar parallax framework.
+• We test our proposed method on large-scale self-
+driving datasets, KITTI [13] and Waymo Open Dataset
+(WOD) [46]. Extensive results demonstrate that our
+method beats the state-of-the-art methods by a large
+margin about 28.8%.
+2. Related Work
+2.1. Monocular Depth Estimation
+Eigen et al. [11] are the ones who first utilize CNN for
+the monocular depth estimation task. They predict depth
+from a single image by combining local and global infor-
+mation. Since then, methods based on neural networks have
+gained significant improvement [12,22,56]. In recent years,
+Vision transformers have been introduced to depth estima-
+tion tasks. Adabin [3] uses a transformer-based architecture
+to divide the depth range into bins. NeWCRFs [60] adopt
+Swin Transformer [27] as encoder and a fully-connected
+Pw
+P'w
+p
+pt
+ps
+p's
+es
+et
+Is
+Os
+Ot
+P
+h
+π
+u
+hc
+w
+p'w
+It
+res
+u'res
+P'
+p't
+Figure 2. The illustration of planar parallax geometry.
+Conditional Random Field(CRF) as decoder, which made a
+considerable improvement. Despite learning from a single
+image, some other work focus on using monocular videos.
+Monocular videos are mainly used by self-supervised mod-
+els [14, 19, 23, 32, 39, 44, 49, 57, 66, 67]. Geometric con-
+straints are built between consecutive frames to reduce the
+dependency on ground-truth depth. We believe that geomet-
+ric information can not only provide supervision in unsuper-
+vised settings but also help supervised methods to achieve
+better performance and generalization.
+2.2. Flow Estimation
+Start from FlowNet series [10, 15], end-to-end optical
+flow networks have shown their superiority in flow estima-
+tion tasks. After that, PWC-Net [45] introduces the pyra-
+mid(P), warping(W), and cost volume(C) into network de-
+sign and significantly improves the performance.
+Mask-
+FlowNet [62] resolves the occluded areas during warping by
+a self-learned occlusion mask. The success of RAFT [47]
+lies in the iterative refinement on the cost volumes. GM-
+Flow [54] first uses a transformer in flow estimation. Along
+with the supervised methods, photometric loss based unsu-
+pervised flow [18,35,52,64] achieves researchers’ attention,
+but there still exists a gap in performance compared with su-
+pervised methods. Optical flow is a fundamental low-level
+task that plays an essential role in many downstream prob-
+lems. We use it as a geometric prior so that the network can
+more efficiently use the geometric information to estimate
+the depth.
+2.3. Planar Parallax Methods
+The planar parallax methods were first proposed in
+the mid-90s in
+[40, 41, 43]. After choosing a reference
+plane, this method decomposes the motion between mul-
+tiple frames into planar homography and residual pixel dis-
+placement from the unaligned pixels [16]. The planar ho-
+mography can be computed by camera z axis translation and
+plane normal vector. The residual pixel displacements can
+2
+
+be obtained by feature matching methods such as optical
+flow. In this way, P+P geometry significantly reduces the
+estimation space. Many applications [8,17,42,58] depends
+on a plane in the scene. For example, in robotics [2,29], the
+height of points from the ground plane is crucial. Moreover,
+Jung et al. [20] propose a method based on planar parallax
+for quantifying image stitching, and Vaish et al. [48] use
+it in camera calibration. A critical problem in these meth-
+ods is finding a suitable reference plane. Naturally, in driv-
+ing scenes, the road plane can easily be extracted from Li-
+DAR points or high-precision maps. Taking these advan-
+tages, Yuan et al. [59] proposed a new solution combining
+traditional planar parallax geometry with a deep neural net-
+work for road environment. Xing et al. [53] uses the dense
+structure information provided by P+P as depth hints. Dur-
+ing our experiments, we found that previous learning-based
+P+P methods do not fully utilize the geometry structure. In
+this paper, we will pursue this goal.
+3. Method
+In this section, we first briefly review planar parallax ge-
+ometry, showing how the method connects 3D structure γ
+with residual image displacement. And then, we elaborate
+on combining this method with deep learning, which enjoys
+the best of two worlds.
+3.1. Planar Parallax Geometry
+For better understanding, we use capital letter to repre-
+sent 3D points, lowercase letter for 2D points, bold font for
+vectors, and matrix in calligraphy. As [16] shown, the ratio
+of height to depth γ plays an important role in modeling the
+homography as a 2D projective transformation. In particu-
+lar,
+γ = h
+z ,
+(1)
+where h and z is the height and depth of a pixel.
+In Fig. 2, we describe the geometry visually.
+Define
+Ps = (x′, y′, z′)T and Pt = (x, y, z)T as the coordinates
+of a point P in source view and target view, separately. Let
+R and T = (tx, ty, tz)T denote the rotation matrix and
+translation vector between the two camera views. The trans-
+formation from Ps to Pt can be written as:
+Pt = RPs + T.
+(2)
+The height above the reference plane π of the point Pt
+can be express as:
+h = hc − ⃗NT Pt,
+(3)
+where ⃗NT is the normal of plane π and hc is the height of
+the camera.
+Let ps = 1
+zKPs, pt = 1
+zKPt and t = KT, where K
+is intrinsic matrix of the camera. The homography matrix
+between the two images can be written as:
+H = K(R + T⃗NT
+hc
+)K−1.
+(4)
+Define pw as the ps warped by homography H,
+pw = Hps
+(5)
+As shown in Fig. 2, ures = pw − pt is the residual flow.
+Following the mathematical derivation in [16, 59], when
+tz ̸= 0, we can obtain:
+ures =
+γ tz
+hc
+1 − γ tz
+hc
+(pt − et),
+(6)
+where et =
+1
+tz t is the epipole in the target view, which
+denotes the point that does not move after the warping. For
+detailed derivation, we refer the readers to the appendix. In
+Fig. 2, P
+′ is a 3D point higher than P. Because γ = h/z,
+for the points with the same depth z, the higher it is, the
+residual flow is larger, equivalently u
+′
+res > ures. The first
+row in Fig. 1 also shows that intuitively. The higher position
+has larger distortion. This is the key geometric clue for 3D
+structure.
+From Eqn. 6, we can conclude that the residual flow al-
+ways moves towards or away from the epipole, and has a
+strong relationship with γ. We can also convert Eqn. 6 to
+γ =
+hc
+tz(1 + pt−et
+ures ).
+(7)
+Except the relationship with residual flow, γ can also pre-
+form 3D reconstruction. Since P can be calculated by an
+inverse projection
+Pt = zK−1pt.
+(8)
+By substituting it into Eqn. 3, we can obtain an important
+formula discussed in proposed Planar Position Embedding.
+⃗NT (K−1pt) = hc − h
+z
+.
+(9)
+Eqn. 9 can be finally transformed into
+z =
+hc
+γ + ⃗NT (K−1pt)
+.
+(10)
+We could use it to convert predicted γ to depth results
+given the plane and camera height above the plane. Com-
+pared with epipolar geometry, the superiority of the P+P ge-
+ometry is twofold: First, as shown in the first row in Fig. 1,
+the road pixel in the image is aligned without disparity after
+warping by road homography. It saves the network capacity
+for estimating the depth of the ground. Second, Eqn. 7 is
+only affected by tz, which removes the dependency on the
+rotation, reducing the error caused by ego-motion noise.
+3
+
+Warped Image
+Target Image
+𝑆
+Self Attention
+Cross Attention
+FFN
+Flow Head
+Score Layer
+Correlation
+Softmax
+Self Attention
+PPE
+Gamma
+Depth
+Transform
+𝐿𝛾
+𝐿𝑑
+Swin-T
+Swin-T
+weight sharing
+Single Frame 
+Branch
+Feature Enhancement
+× 𝐿 layers
+Weighted Sum
+Embedding Layer
+Figure 3. Overview of the proposed Planar Parallax Network. Given two plane-aligned images, we first extract features by a swin-tiny
+backbone. And then we divide two streams, a flow branch and a single frame branch. In flow branch, we follow the Feature Enhancement
+module and Flow Head in GMflow [54]. The Flow Head is only used in flow pretrain. The Single Frame Branch is a simple conv net and
+fused with flow branch by weighted sum. The network is supervised by gamma loss Lγ and depth loss Ld.
+3.2. Planar Parallax Network
+Flow Network Pre-training As shown in Eqn. 7, given the
+height of the camera and the translation along the forward
+axis, γ can be described by residual flow based on plane-
+aligned images. The pixel fall on the plane has zero height
+which leads to γ = 0, ures = 0, and the flow can recon-
+struct the height and depth. To enhance the accuracy of
+residual flow, we try to introduce geometric prior by pre-
+training the network by flow estimation task [54,62]. After
+that, the γ prediction becomes a much easier assignment.
+As in Tab. 3, the flow pre-training brings significant im-
+provement.
+Prediction for Dynamic Objects For moving objects that
+violate the underlying static scene assumption in Sec. 3.1,
+we follow the recent work DfM [51] that introduces an ad-
+ditional component. It utilizes only the reference image as
+input and predicts γ, which mainly benefits from the train-
+ing data and network’s generalization power. This branch is
+fused into the primary branch as:
+S = σ[φ(Fm, Fs)],
+(11)
+F = S ◦ Fm + (1 − S) ◦ Fs,
+(12)
+where σ is the sigmoid function. ◦ is element-wise multi-
+plication. φ is a simple convolutional layer, which outputs
+fusion score S guiding the fusion of the single-frame feature
+Fs and multi-frame feature Fm.
+Planar Position Embedding (PPE) In P+P methods, there
+are two parameters the camera’s intrinsic K and normal vec-
+tor of the plane ⃗NT . The network should be aware these
+two variables. Inspired by position embedding from trans-
+former models, we propose Planar Position Embedding,
+which combines these two into one formula ⃗NT (K−1p).
+As shown in Eqn. 9, the PPE is the projection of the point
+in the normalized image plane on the normal direction of
+the reference plane. We show the visualization of PPE in
+Fig. 3. PPE distinctly describes the trend of change when
+the plane is tilted in the camera. We use it as the pixel’s
+position related to the reference plane and build embedding
+as follows:
+F = φ(E)
+(13)
+Eij = ⃗NT (K−1pij)
+(14)
+where φ is a simple convolutional network. It lets the net-
+work aware the relative position of the plane, suppress-
+ing some absurd errors. We investigate its effectiveness in
+Sec. 4.4.
+4
+
+Overall Structure Combining the method mentioned
+above, we propose a new framework named PPNet.
+As
+shown in Fig. 3, the network takes two consecutive im-
+ages It−1 and It as input.
+image It−1 is warped using
+road plane homography. The network outputs γ map of
+image It, which is then transformed into depth map us-
+ing Eqn. 10. We adopt the feature extraction and feature
+enhancement component introduced in GMFlow [54] and
+leave the feature matching and flow propagation only for
+flow pre-training.
+We replace the CNN backbone in GMFlow with a Swin
+Transformer [27] for greater model capacity. The 1
+8 down-
+sampled feature map is fed into feature enhancement trans-
+former, where the P+P geometry can be analyzed. The
+1
+32
+downsampled feature map is used for single frame estima-
+tion, which is then fused with the output of the feature en-
+hancement transformer. After the fusion, planar position
+embedding will be introduced. The final feature is upsam-
+pled to the original size to output γ.
+Training Loss. Following previous work [3,11,22,59,60],
+we use a L1 loss to supervise γ and a Scale-Invariant Log-
+arithmic (SILog) loss to supervise the depth obtained by γ
+using Eqn. 10. If the total number of pixels with ground-
+truth is N, the loss for γ can be define as
+Lγ = 1
+N
+�
+i
+|γi − γ∗
+i |,
+(15)
+where γi and γ∗
+i are the predicted γ value and correspond-
+ing ground-truth.
+The logarithm difference is defined as
+∆di = log di − log d∗
+i ,
+(16)
+where di and d∗
+i are the predicted depth value obtained by
+γ using Eqn. 10 and its corresponding ground-truth depth
+value. Then the depth loss is defined as:
+Ld = α
+�
+1
+N
+�
+i
+∆d2
+i − λ
+N 2 (
+�
+i
+∆di)2,
+(17)
+where λ is a variance minimizing factor, and α is a scale
+constant. Following the previous works [22,60], we set λ =
+0.85 and α = 10.
+The total loss function is defined as the summation of the
+loss for γ and depth with weight wγ and wd:
+L = wγLγ + wdLd.
+(18)
+Here we set wγ = 1 and wd = 10−2 since the depth
+produced by Eqn. 10 may be very unstable. Also, the net-
+work should focus more on the geometric advantage that γ
+brings.
+4. Experiments
+In this Section, we first introduce two autonomous driv-
+ing datasets in Sec. 4.1, KITTI and WOD, and describe how
+we build the data our model needs. Then we provide the
+implementation details of our method in Sec. 4.2. Sec. 4.3
+demonstrates that our method dramatically exceeds the pre-
+vious state-of-the-art methods in both datasets. We conduct
+specific ablation studies in Sec. 4.4. On the one hand, we
+prove our main point that the flow pre-trained model effec-
+tively helps the model using the geometric prior. On the
+other hand, it shows that the improvements we proposed
+are practical and critical.
+4.1. Datasets
+We
+use
+KITTI
+dataset
+[13]
+and
+Waymo
+Open
+Dataset [46] to evaluate the performance of the proposed
+network.
+KITTI. KITTI dataset is one of the most popular bench-
+mark in Monocular Depth Estimation. We use the data split
+proposed by Eigen et al. [11], which contains 23488 train-
+ing samples and 697 testing samples. We reproject all the
+points on the ground-truth depth map back to 3D space and
+extract the road plane using RANSAC algorithm. As in pre-
+vious work, the homography transformation is calculated
+using odometry data provided by KITTI. Moreover, we fix
+some inaccurate pose matrix using point-to-plane ICP algo-
+rithm [7]. For the reference plane, we adopt two different
+settings to show the universality of our work:
+• Estimated Plane(EP). In this setting, the road planes
+are extracted using the RANSAC algorithm to fit a
+flat plane in the scene. For the images without a road
+plane, we use the mean plane method below.
+• Mean Plane(MP). In this setting, We demonstrate the
+capabilities of our model without the accurate planes.
+The mean plane is acquired by computing the mean
+plane normal of the estimated plane from all the frames
+in the dataset. Note that if we have accurate extrinsic
+parameters of camera w.r.t. the road plane, we could
+also approximately extract planes using methods like
+IPM [33]. However, it is not provided accruractly in
+the dataset.
+Waymo Open Dataset. Since the RP2-Waymo dataset [59]
+remains unpublished, to compare with RPANet [59], we re-
+produced the dataset following it. The reproduced dataset
+contains 12894 training samples and 1345 test samples.
+WOD does not have dense depth supervision. The ground-
+truth γ is built by the sparse observations from LiDAR. Al-
+though the experiments on WOD show the generalization
+of methods, we also argue that without a deliberate process-
+ing, the ground-truth depth from LiDAR may also lead to
+5
+
+Method
+Backbone
+Abs Rel ↓
+Sq Rel ↓
+RMSE ↓
+RMSE log ↓
+δ1 ↑
+δ2 ↑
+δ3 ↑
+Params
+Eigen et al. [11]
+-
+0.190
+1.515
+7.156
+0.270
+0.692
+0.899
+0.967
+83 M
+DORN [12]
+ResNet-101
+0.072
+0.307
+2.727
+0.120
+0.932
+0.984
+0.995
+100 M
+BTS [22]
+ResNext-101
+0.059
+0.241
+2.756
+0.096
+0.956
+0.993
+0.998
+113 M
+DPT [38]
+VIT-Hybrid
+0.062
+0.222
+2.575
+0.092
+0.959
+0.995
+0.999
+123 M
+Adabin [3]
+EfficientNet-B5
+0.058
+0.190
+2.360
+0.088
+0.964
+0.995
+0.999
+78 M
+NeW CRFs [60]
+Swin-Large
+0.052
+0.155
+2.129
+0.079
+0.974
+0.997
+0.999
+270 M
+Ours(MP)
+Swin-Tiny
+0.044
+0.127
+1.986
+0.069
+0.981
+0.997
+0.999
+52 M
+Ours(EP)
+Swin-Tiny
+0.037
+0.109
+1.815
+0.062
+0.983
+0.997
+0.999
+52 M
+Table 1. Quantitative results on the Eigen split of KITTI dataset. The best results are in bold and second best are underlined.
+Method
+Height
+Abs Rel ↓
+Sq Rel ↓
+RMSE ↓
+RMSE log ↓
+δ1 ↑
+δ2 ↑
+δ3 ↑
+RPANet [59]
+< 1m
+0.036
+0.198
+2.707
+0.080
+0.974
+0.992
+0.997
+BTS [22]
+< 1m
+0.044
+0.166
+2.383
+0.075
+0.980
+0.995
+0.998
+NeW CRFs [60]
+< 1m
+0.043
+0.155
+2.321
+0.072
+0.981
+0.996
+0.999
+Ours(EP)
+< 1m
+0.029
+0.131
+2.200
+0.065
+0.984
+0.995
+0.998
+RPANet [59]
+-
+0.086
+1.089
+5.623
+0.187
+0.903
+0.968
+0.987
+BTS [22]
+-
+0.071
+0.531
+4.105
+0.119
+0.939
+0.984
+0.995
+NeW CRFs [60]
+-
+0.067
+0.459
+3.866
+0.112
+0.945
+0.987
+0.996
+Ours(EP)
+-
+0.056
+0.450
+3.853
+0.108
+0.950
+0.987
+0.995
+Table 2. Quantitative results on the Waymo Open Dataset.
+Figure 4. Mismatch examples of Waymo Open Dataset. Left Up:
+motion distortion. Right Up: noise of high reflectance. Left Bot-
+tom: rainy noise. Right Bottom: unknown noise.
+some unexpected errors. As shown in Fig. 4, objects with
+high reflectance or motion distortion will mismatch the ob-
+servations from the image and LiDAR.
+4.2. Implementation Details
+We implement the proposed network using Pytorch [37].
+AdamW optimizer [28] with a weight decay of 10−2 is
+adopted. All experiments are preformed on Nvidia RTX
+3090 GPUs. Following
+[22], the learning rate decrease
+from 10−4 to 10−5 using polynomial decay with power
+p = 0.9. Our model is trained for 20 epochs with a to-
+tal batch size of 8. We use augmentation techniques such as
+horizontal flipping and brightness jittering. Moreover, since
+the P+P geometry is built on the condition tz ̸= 0, we ran-
+domly replace the warped image It−1 with image It to im-
+prove the robustness to the static scene. Our model is first
+pre-trained on flow estimation task. We follow the train-
+ing strategy of KITTI dataset in GMFlow [54], the model
+is first traind on FlyingChairs(Chairs) [10] and FlyingTh-
+ings3D (Things) [34] datasets, then fine-tuned on Sintel [6]
+and KITTI [36] datasets.
+4.3. Comparisons with the state-of-the-art Depth
+Depth results on KITTI dataset In Tab. 1, we report the re-
+sults of depth estimation on KITTI, in which our model sur-
+passes the previous state-of-the-art model by a significant
+margin on all the metrics. Notably, although we can easily
+get the estimated plane(EP) during the autonomous driving
+scenes, we also show the results with the mean plane(MP),
+which does not rely on any online estimation. With some
+loss of accuracy by the estimated plane, the improvement is
+still considerable. Fig. 5 illustrates the quantitative results.
+As shown in the error map, we highlight the advantages of
+our method in estimating obstacles. Since there are priors
+provided by the ground, the error of vertical and static ob-
+jects are significantly improved.
+Depth results on Waymo Open Dataset As for Waymo
+Open Dataset, we reproduce the RPANet following [59]
+6
+
+NismettaInput Image
+BTS [22]
+NeW CRFs [60]
+Ours
+Figure 5. Qualitative results on the Eigen split of KITTI dataset. For each sample, the first column shows the target image and the flow
+estimated by the pre-trained optical flow model between the two plane-aligned images. The rest columns each shows the predicted depth
+map and the corresponding error map for a model. Blue represents smaller error, while red represents larger error.
+and train BTS [22], NeW CRFs [60] with official open-
+sourced code for comparison1.
+In Tab. 2, we show the
+results with height < 1m and full range. Compared with
+the results in KITTI, PPNetstill shows remarkable improve-
+ment in Abs Rel, but the improvement gap in Sq Rel has
+been narrowed. That means our model still has a higher
+accuracy, but there are more pixels with larger errors in all
+methods. This is because WOD includes some difficult data
+on nights or rainy days. Moreover, there are many slow-
+driving scenarios which violate the motion assumption in
+Eqn. 6. Despite all this, our methods show the priority in
+height < 1m, benefiting from the plane prior.
+4.4. Ablation Study
+Effectiveness of flow pre-training To better understand
+the benefit flow pre-training brings, we remove the single
+frame branch, Planar Position Embedding, and the random
+data augmentation so that no extra single frame information
+would affect the result in this ablation study. After remov-
+ing these components, we also add a condition height< 1m
+to highlight the influence caused by the prior of plane. In
+1http://github.com/cleinc/bts, http://github.
+com/aliyun/NeWCRFs
+Target
+Frame
+Warp
+Pretrain
+Abs Rel ↓
+RMSE ↓
+Flow
+2
+N
+ImageNet
+0.160
+5.661
+Depth
+1
+N
+ImageNet
+0.059
+2.059
+Gamma(γ)
+1
+N
+-
+0.055
+2.271
+Gamma(γ)
+2
+N
+-
+0.054
+2.231
+Gamma(γ)
+2
+Y
+-
+0.054
+2.191
+Gamma(γ)
+1
+N
+ImageNet
+0.047
+1.985
+Gamma(γ)
+2
+N
+ImageNet
+0.046
+1.944
+Gamma(γ)
+2
+Y
+ImageNet
+0.045
+1.927
+Gamma(γ)
+1
+N
+Flow
+0.044
+1.965
+Gamma(γ)
+2
+N
+Flow
+0.039
+1.755
+Gamma(γ)
+2
+Y
+Flow
+0.035
+1.460
+Table 3.
+Ablation studies for flow pre-training. The target flow
+means we use the flow prediction and compute γ by Eqn. 7. Frame
+indicates to number of consecutive frames we use. Warp denotes
+whether to warp with planar homography. All results are under
+condition height< 1m.
+Tab. 3, we first show the pure geometry methods. We com-
+pute γ by Eqn. 7 with flow prediction from GMFlow. This
+method merely does not work because dynamic objects do
+not meet the hypothesis and some pixels near to the epipole
+are too sensitive to flow’s precision. Furthermore, the com-
+7
+
+Figure 6. Row 1: Original images. Row 2: Predicted depth map
+without PPE. Row 3: Predicted depth map with PPE. Row 4: Error
+map without PPE. Row 5: Error map with PPE.
+Method
+Abs Rel ↓
+Sq Rel ↓
+RMSE ↓
+baseline
+0.073
+0.411
+3.378
++FP
+0.044
+0.216
+2.165
++PPE
+0.040
+0.145
+2.013
++SFB
+0.037
+0.117
+1.878
++DL
+0.037
+0.109
+1.815
+Table 4.
+Ablation studies for components. The components is
+added incrementally. Flow pretrain is shown as FP. SFB means
+single frame branch. DL is the auxiliary depth loss described in
+Eqn. 17. PPE is proposed Planar Position Embedding.
+parison between depth and γ shows the superiority of γ
+prediction.
+As in Eqn. 10, γ is independent of intrinsic
+K, which improves its generalization. Above all, we con-
+duct experiments in different pre-training, from scratch, Im-
+ageNet and optical flow. The results indicate that the model
+with ImageNet pre-training can not take advantage of the
+consecutive frame shown by the similar performance be-
+tween frame one or two. On the contrary, the flow pre-
+training significantly improves the results. Finally, to fig-
+ure out the influence of planar prior, we show the model
+without warping, which only uses the epipolar geometry.
+The results are still worse than our proposed method, which
+means the planar prior plays an essential role in our superior
+results.
+Effect of each component. In Tab. 4, we unfold the pro-
+posed model module by module to understand how each
+component affects our results. We first validate the effec-
+tiveness of the essential idea, the flow pre-training. It almost
+halves the error. Then, Planar Position Embedding reduce
+the Sq Rel by more than 25%. Specifically, its influence is
+Figure 7. Row 1: Homography aligned images pairs, the first sam-
+ple contains moving cars, the second sample does not have ego
+motion. Row 2: Predicted depth map without SFB. Row 3: Pre-
+dicted depth map with SFB. Row 4: Error map without SFB. Row
+5: Error map with SFB.
+more intuitive in Fig. 6. Planar Position Embedding con-
+tains the pixels’ position related to the ground plane, which
+can restrain some unreasonable errors caused by road with
+different slope or moving objects. As shown in Fig. 7, a
+single frame branch improves the performance on dynamic
+objects and static frames. Lastly, adding additional depth
+supervision makes the network learn the depth information
+directly, leading the error to the final level.
+5. Conclusions and Future Work
+This paper presents Planar Parallax Network, a simple
+but effective depth estimation framework based on planar
+parallax geometry. By deliberately analyzing geometric in-
+formation’s effectiveness, our method introduces the flow
+pretrain to make the network learning start from an initial-
+ization well-tuned by geometric prior. Then we handle the
+inherent weakness of the planar parallax pipeline by single
+frame estimation and Planar Position Embedding. Compre-
+hensive experiments on KITTI and Waymo Open Dataset
+demonstrate PPNetsurpsasses the previous SOTA methods
+by a significant margin.
+For future work, low-cost flow supervision will be a po-
+tential topic. Many unsupervised flow methods can be joint
+training in our framework. Furthermore, we would like to
+extend our method to multi-frame observation, which can
+provide more geometric information.
+Moreover, we are
+also interested in integrating our method into the real au-
+tonomous driving perception system.
+8
+
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+10
+
+A. Planar Parallax Geometry
+We provide a complete derivation process in this section.
+As in the main body, we use capital letter to represent 3D
+points, lowercase letter for 2D points, bold font for vectors,
+and matrix in calligraphy.
+The ratio of height to depth γ is defined as:
+γ = h
+z ,
+(19)
+where h and z is the height and depth of a pixel.
+Define Ps = (x′, y′, z′)T and Pt = (x, y, z)T as the
+coordinates of a point P in source view and target view,
+separately. Let R and T = (tx, ty, tz)T denote the rota-
+tion matrix and translation vector between the two camera
+views. The transformation from Ps to Pt can be written as:
+Pt = RPs + T.
+(20)
+The height above the reference plane π of the point P
+can be express as:
+h = hc − ⃗NT P,
+(21)
+where ⃗NT is the normal of plane π and hc is the height of
+the camera. Eqn. 21 can be transformed into:
+h + ⃗NT P
+hc
+= 1.
+(22)
+By multiply T by 1 in Eqn. 20, we can obtain
+Pt = RPs + Th + ⃗NT Ps
+hc
+= (R + T⃗NT
+hc
+)Ps + h
+hc
+T
+(23)
+Let ps = 1
+z′ KPs, pt = 1
+zKPt and t = KT, where K is
+intrinsic matrix of the camera. Then we can obtain
+zK−1pt = (R + T⃗NT
+hc
+)z′K−1ps + h
+hc
+T
+(24)
+By mutiply 1
+z′ K on both sides, we have:
+z
+z′ pt = K(R + T⃗NT
+hc
+)K−1ps +
+h
+hcz′ t.
+(25)
+With the homography matrix between the two images
+written as:
+H = K(R + T⃗NT
+hc
+)K−1,
+(26)
+Eqn. 25 can be reformulated as
+z
+z′ pt = Hps +
+h
+hcz′ t.
+(27)
+By considering the z-axis of both sides, we have:
+z
+z′ = H3ps + htz
+hcz′ ,
+(28)
+where H3 denote the third row of homography matrix H
+Note that, the z-axis of ps and pt is 1. Scaling both sides
+by their z-axis, we can obtain
+pt =
+Hps +
+h
+hcz′ t
+H3ps + htz
+hcz′
+= Hps
+H3ps
+− Hps
+H3ps
++
+Hps +
+h
+hcz′ t
+H3ps + htz
+hcz′
+= Hps
+H3ps
+−
+htz
+hcz′
+(H3ps + htz
+hcz′ )
+Hps
+H3ps
++
+h
+hcz′ t
+H3ps + htz
+hcz′
+= Hps
+H3ps
+− htz
+zhc
+Hps
+H3ps
++
+h
+hcz t.
+(29)
+With epipole et =
+1
+tz t, γ = h
+z , ps warped by homogra-
+phy pw =
+Hps
+H3ps , when tz = 0, we have
+pt = pw +
+h
+hcz t.
+(30)
+When tz ̸= 0, we have
+pt = pw − γ tz
+hc
+(pw − et).
+(31)
+Then we can obtain
+pw − pt = γ tz
+hc
+(pw − et),
+(32)
+which can also be converted to
+pw − pt = γ tz
+hc
+(pw − pt + pt − et)
+(33)
+(1 − γ tz
+hc
+)(pw − pt) = γ tz
+hc
+(pt − et)
+(34)
+pw − pt =
+γ tz
+hc
+1 − γ tz
+hc
+(pt − et)
+(35)
+Now we get the relationship between ures = pw − pt and
+γ.
+B. More Qualitative Results
+In Fig. 8, we show more qualitative results of BTS [22],
+NeW CRFs [60] and our method. As shown in the error
+maps, the result have improved significantly.
+In Fig. 9, we show an example of how each component
+affects the result. By adding flow pre-training, the perfor-
+mance is improved significantly on static scenes but wors-
+ened on dynamic objects that violate the static assumption
+11
+
+in planar parallax geometry. It suggests that, without flow
+pre-training, models may not take advantage of the consecu-
+tive frame. More examples are shown in Fig.10. By adding
+Planar Position Embedding, the unreasonable errors have
+been restrained. Finally, the single frame branch and depth
+supervision improves the performance on dynamic objects
+and leads the error to the final level.
+12
+
+Input Image
+BTS [22]
+NeW CRFs [60]
+Ours
+Figure 8. Qualitative results on the Eigen split of KITTI dataset. For each sample, the first column shows the target image and the flow
+estimated by the pre-trained optical flow model between the two plane-aligned images. The rest columns each shows the predicted depth
+map and the corresponding error map for a model. Blue represents smaller error, while red represents larger error.
+13
+
+皖9PHK·C1338888mobel ms:
+HieFer KefsInput Image
+Baseline
++FP
++PPE
++SFB
++DL
+Figure 9. Qualitative results for components. The components is added incrementally. Flow pretrain is shown as FP. SFB means single
+frame branch. DL is the depth loss. PPE is the proposed Planar Position Embedding. The first row shows the target image and the
+plane-aligned image pairs. The rest rows shows the predicted depth map and the corresponding error map separately.
+14
+
+Input Image
+Baseline
++FP
+Input Image
+Baseline
++FP
+Input Image
+Baseline
++FP
+Figure 10. Comparison between baseline and +FP. For each sample, the first row shows the target image and the plane-aligned image pairs.
+The rest rows shows the predicted depth map and the corresponding error map separately.
+15
+
+VerwaltenerwaltenERAEL

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+Optimally-Weighted Estimators of the Maximum
+Mean Discrepancy for Likelihood-Free Inference
+Ayush Bharti 1
+Masha Naslidnyk 2
+Oscar Key 2
+Samuel Kaski 1,3
+François-Xavier Briol 2,4
+1 Department of Computer Science, Aalto University, Finland
+2 Department of Statistical Science, University College London, London, UK
+3 Department of Computer Science, University of Manchester, UK
+4 The Alan Turing Institute, UK
+Likelihood-free inference methods typically make use of a distance between simulated and real
+data. A common example is the maximum mean discrepancy (MMD), which has previously been
+used for approximate Bayesian computation, minimum distance estimation, generalised Bayesian
+inference, and within the nonparametric learning framework. The MMD is commonly estimated
+at a root-m rate, where m is the number of simulated samples. This can lead to significant
+computational challenges since a large m is required to obtain an accurate estimate, which is
+crucial for parameter estimation. In this paper, we propose a novel estimator for the MMD
+with significantly improved sample complexity. The estimator is particularly well suited for
+computationally expensive smooth simulators with low- to mid-dimensional inputs. This claim
+is supported through both theoretical results and an extensive simulation study on benchmark
+simulators.
+1. Introduction
+Many domains of science, medicine and engineering use our mechanistic understanding of real-world
+phenomena to create simulators that can represent system behaviour in different circumstances. Such
+simulator-based models define a stochastic procedure that can generate (possibly complex) synthetic data-
+sets, and are widely used in fields such as population genetics Beaumont (2010), ecology Wood (2010),
+astronomy Cameron and Pettitt (2012); Akeret et al. (2015), epidemiology Kypraios et al. (2017), atmospheric
+contamination Kopka et al. (2016), radio propagation Bharti et al. (2022a), and agent-based modelling
+Jennings (1999). However, the ease of simulating data from the model comes at the cost of an intractable
+likelihood function, rendering most standard statistical inference methods inapplicable to such models. To
+solve this issue, a host of likelihood-free inference methods have been developed that circumvent the need to
+evaluate the likelihood or its derivatives, see Lintusaari et al. (2017); Cranmer et al. (2020) for an overview.
+A common approach for likelihood-free inference involves comparing simulated observations from the
+model and the observed data, with respect to some notion of distance. Accurately estimating the distance is
+essential for inference but doing so usually requires simulating large amounts of synthetic data. This can be
+a computational bottleneck, especially for expensive simulators, which in the most extreme cases can take up
+1
+arXiv:2301.11674v1  [stat.ME]  27 Jan 2023
+
+equally-weighted
+True
+optimally-weighted 
+(ours) 
+Figure 1.: Estimating the MMD requires approximating the embedding µk,Pθ of the model Pθ in a reproducing
+kernel Hilbert space Hk. The classical approach consists of doing this from m equally-weighted
+independent samples from Pθ (denoted µEW
+k,Pm
+θ ), but we show in this paper that it is possible to
+improve this estimator by using optimally-weighted samples (denoted µOW
+k,Pm
+θ ).
+to hundreds or thousands of CPU hours per simulation; see Niederer et al. (2019) for an example in cardiac
+modelling. Other examples include tsunami models based on shallow water equations that require several
+GPU hours per run Behrens and Dias (2015), runaway electron analysis models for nuclear fusion devices
+that require 24 CPU hours per run Hoppe et al. (2021), and models of large-scale wind farms that require
+100 CPU hours per run Kirby et al. (2022). Naturally, the discrepancies popular for likelihood-free inference
+are those which can be efficiently estimated given samples from two distributions, such as the KL divergence
+Jiang (2018), Wasserstein distance Peyré and Cuturi (2019); Bernton et al. (2019), Sinkhorn divergence
+Genevay et al. (2018), energy distance Nguyen et al. (2020), classification accuracy Gutmann et al. (2017),
+or the maximum mean discrepancy, the latter of which is the topic of this paper. Here, “efficiently estimated”
+is defined in terms of sample complexity, that is, how many samples from the distributions we need to
+estimate a distance. The faster this error goes to zero with the number of samples, the less we need to
+simulate from the model, and hence, the smaller the computational cost.
+We focus on the maximum mean discrepancy (MMD) Gretton et al. (2006, 2012), a probability metric
+which measures the distance between distributions through the distance between their embeddings in a
+reproducing kernel Hilbert space; see Figure 1 for an illustration. A number of advantages of this distance
+are commonly put forward in the literature: (i) it has relatively low sample complexity when compared to its
+alternatives listed above, (ii) it has desirable statistical properties, such as leading to consistent and robust
+estimators, (iii) it is applicable on any data-type for which a kernel can be defined, and does not require
+hand-crafted summary statistics. Due to these attractive properties, the MMD has been used in a range of
+frameworks for likelihood-free inference, including for approximate Bayesian computation (ABC) Park et al.
+(2015); Mitrovic et al. (2016); Kajihara et al. (2018); Bharti et al. (2022a); Legramanti et al. (2022), for
+minimum distance estimation (MDE) Briol et al. (2019a); Chérief-Abdellatif and Alquier (2021); Alquier
+and Gerber (2021); Niu et al. (2021); Key et al. (2021), for generalised Bayesian inference Chérief-Abdellatif
+and Alquier (2020); Pacchiardi and Dutta (2021), for Bayesian nonparametric learning Dellaporta et al.
+(2022), and for training generative adversarial networks Dziugaite et al. (2015); Li et al. (2015, 2017a);
+Bińkowski et al. (2018).
+In this paper, we do not revisit the question of whether the MMD is the best choice of distance for
+a particular problem. Instead, we assume that the MMD has been chosen, and focus on constructing
+estimators with strong sample complexity for this distance. The most common estimators for the MMD are
+U-statistic or V-statistic estimators, and these have sample complexity of O(m− 1
+2 ), under mild conditions
+Briol et al. (2019a), where m is the number of samples. In recent work, Niu et al. (2021) showed that this
+can be improved to O(m−1+ϵ) for any ϵ > 0 through the use of a V-statistic estimator and randomised
+2
+
+quasi-Monte Carlo (RQMC) sampling. This significant improvement does come at the cost of restrictive
+assumptions — the simulator must be written in a form where the inputs are uniform random variables,
+and must satisfy stringent smoothness conditions which are difficult to verify in practice.
+In this paper, we propose a novel set of optimally-weighted estimators with sample complexity of
+O(m− νc
+s − 1
+2 ) where s is the dimension of the base space and νc is a parameter depending on the smoothness
+of the kernel and the simulator. This leads to significantly improved sample complexity against both U-
+or V-statistic and independent samples for any νc, and against RQMC when νc > s/2. Additionally, the
+optimality of the weights guarantees that even if this condition is not satisfied, the order of the sample
+complexity is still at least as good as that for existing estimators.
+The remainder of the paper is structured as follows. Section 2 recalls existing estimators for the MMD, and
+how these are used in likelihood-free inference. Section 3 presents our estimators, and Section 4 provides a
+theoretical analysis of their sample complexity. Finally, Section 5 demonstrates strong empirical performance
+on a range of simulators, and Section 6 discusses future research.
+2. Background
+Throughout the paper, X will denote some set, and P(X) will be the set of all Borel probability measures
+on X.
+Likelihood-free inference
+We consider the classic parameter estimation problem, where we assume that
+we observe some independent and identically distributed (iid) realisations {xi}n
+i=1 ⊆ X from some data-
+generating mechanism Q ∈ P(X). Given {xi}n
+i=1 and a parametric family of distributions {Pθ : θ ∈ Θ} ⊂
+P(X) (i.e. the model) with parameter space Θ, we are interested in recovering the parameter value θ∗ ∈ Θ
+such that Pθ∗ is either equal, or in some sense closest, to Q.
+The challenge in likelihood-free inference is that the likelihood associated with Pθ is intractable, meaning
+it cannot be evaluated pointwise. This prevents the use of classical methods such as maximum likelihood
+estimation or (exact) Bayesian inference. Instead, we assume that we are able to simulate iid realisations
+from Pθ, and such models are hence called generative models or simulator-based models. Such models are
+characterised through their generative process, a pair (Gθ, U) consisting of a simple distribution U (such as
+a multivariate Gaussian or uniform distribution) on a space U and a map Gθ : U → X called the generator
+or simulator. We will call U a base measure and U the base space, and consider U ⊂ Rs and X ⊆ Rd. To
+sample y ∼ Pθ, one can first sample u ∼ U, then apply the generator y = Gθ(u). To perform parameter
+estimation for these models, it is common to repeatedly sample simulated data from the model for different
+parameter values and compare them to {xi}n
+i=1 using a distance. We now recall the distance which will be
+the focus of this paper.
+Maximum mean discrepancy (MMD)
+Let Hk be a reproducible kernel Hilbert space (RKHS) associated
+with the symmetric and positive definite function k : X × X → R Berlinet and Thomas-Agnan (2004),
+called a reproducing kernel, and denote by ∥ · ∥Hk and ⟨·, ·⟩Hk the corresponding norm and inner product.
+Additionally, let Pk(X) := {P ∈ P(X) :
+�
+X
+�
+k(x, x)P(dx) < ∞}; whenever k is bounded, Pk(X) = P(X).
+As illustrated in the sketch in Figure 1, any distribution P ∈ Pk(X) can be mapped into Hk via its kernel
+mean embedding, defined as µk,P =
+�
+X k(·, x)P(dx). Then, the MMD between P and Q is the distance
+between their embeddings in Hk:
+MMDk(P, Q) = ∥µk,P − µk,Q∥Hk,
+(1)
+see Muandet et al. (2017) for a review. Alternatively, the MMD can also be expressed as MMDk(P, Q) =
+sup∥f∥Hk≤1
+���
+X f(x)P(dx) −
+�
+X f(x)Q(dx)
+�� , where the supremum is taken over all the functions in the unit-
+3
+
+ball of the RKHS Hk. Whenever k is a characteristic kernel, the MMD is a probability metric, meaning that
+MMDk(P, Q) = 0 if and only if P = Q. This condition is satisfied for kernels including the squared-exponential
+(SE) kSE(x, y) = η exp(−∥x−y∥2
+2/l2), the Matérn kν(x, y) =
+η
+Γ(ν)2ν−1 (
+√
+2ν
+l ∥x−y∥2)νKν(
+√
+2ν
+l ∥x−y∥2), where
+Kν is the modified Bessel function of the second kind, and the inverse-multiquadric kernels on X = Rd
+Sriperumbudur et al. (2010). Matérn kernels are of particular interest: the order parameter ν uniquely
+determines the smoothness of Hk, and for half-integer orders ν ∈ {1
+2, 3
+2, . . . }, the kernel kν can be written as
+a product of an exponential and a polynomial of order ⌊ν⌋ (Rasmussen and Williams, 2006).
+Unfortunately, the expression in (1) usually cannot be computed directly since µk,P will not be available
+in closed form outside of a limited number of (k, P) pairs. Instead, using the reproducing property (i.e.
+f(x) = ⟨f, k(·, x)⟩Hk ∀f ∈ Hk), we can write
+MMD2
+k(P, Q) =
+�
+X
+�
+X
+k(x, y)P(dx)P(dy) − 2
+�
+X
+�
+X
+k(x, y)P(dx)Q(dy) +
+�
+X
+�
+X
+k(x, y)Q(dx)Q(dy).
+(2)
+This expression is convenient to work with as it can be estimated through approximations of the integrals.
+Let {yi}m
+i=1 ∼ P, {xi}n
+i=1 ∼ Q and let Pm = 1
+m
+�m
+j=1 δyj and Qn = 1
+n
+�n
+i=1 δxi, where δxi is a Dirac measure
+at xi. The squared-MMD can be approximated through a V-statistic as
+MMD2
+k(Pm, Qn) = 1
+m2
+m
+�
+i,j=1
+k(yi, yj) −
+2
+nm
+n
+�
+i=1
+m
+�
+j=1
+k(xi, yj) + 1
+n2
+n
+�
+i,j=1
+k(xi, xj).
+This is equivalent to approximating µk,P using µEW
+k,Pm(x) =
+1
+m
+�m
+i=1 k(x, xi). Alternatively, one can use
+an unbiased U-statistic approximation Gretton et al. (2012). Both of these estimates can be calculated
+straightforwardly via evaluations of the kernel k at a computational cost O(m2 + mn + n2).
+Likelihood-free inference with the MMD
+The MMD has been used within a range of frameworks. In a
+frequentist setting, the MMD was proposed for minimum distance estimation by Briol et al. (2019a):
+ˆθn = arg min
+θ∈Θ
+MMD2
+k(Pθ, Qn).
+(3)
+In practice, the minimiser is computed through an optimisation algorithm, which requires evaluations of the
+squared-MMD or of its gradient. Such evaluations are intractable, but any estimator can be used within a
+stochastic optimisation algorithm. Similar optimisation problems and stochastic approximations also arise
+when using the MMD for generative adversarial networks Dziugaite et al. (2015); Li et al. (2015) and for
+nonparametric learning Dellaporta et al. (2022).
+In a Bayesian setting, the MMD has been used to create several pseudo-posteriors by updating a prior
+distribution p on Θ using data. For example, the K2-ABC posterior of Park et al. (2015) is a pseudo-posterior
+of the form:
+pABC(θ|x1 . . . , xn) ∝
+�
+· · ·
+�
+m
+�
+j=1
+1{MMD2
+k(Pθ,Qn)<ε}(θ)p(yj|θ)p(θ)dy1, . . . , dym.
+(4)
+where the indicator function 1{A} is equal to 1 if event A holds. Here, the MMD is used to determine
+whether a particular instance of the parametric model is within an ε distance from the data. The K2-ABC
+algorithm approximates this pseudo-posterior through sampling of the model Pθ which leads to the use of
+an estimator of the squared-MMD.
+Finally, the MMD has also been used for generalised Bayesian inference, where it is used to construct the
+MMD-Bayes posterior Chérief-Abdellatif and Alquier (2020)
+pGBI(θ|x1 . . . , xn) ∝ exp(−MMD2
+k(Pθ, Qn))p(θ).
+4
+
+Once again, this pseudo-posterior is intractable, but it can be approximated through pseudo-marginal
+MCMC, in which case an unbiased estimator is used in place of the squared-MMD Pacchiardi and Dutta
+(2021).
+Sample complexity of MMD estimators
+As highlighted above, the performance of these likelihood-free
+inference methods relies heavily on how accurately we can estimate the MMD using samples; that is, how
+fast our estimator approaches MMDk(Pθ, Q) as a function of n and m, the number of observed and simulated
+data points, respectively. Let �
+MMDk(Pm
+θ , Qn) be any estimator of the MMD based on m simulated data
+points. Using the triangle inequality, this error can be decomposed as follows:
+|MMDk(Pθ, Q) − �
+MMDk(Pm
+θ , Qn)| ≤ |MMDk(Pθ, Q) − MMDk(Pθ, Qn)|
++ |MMDk(Pθ, Qn) − �
+MMDk(Pm
+θ , Qn)|
+(5)
+where the first term describes the approximation error due to having a finite number of data points n, and
+the second term describes the error due to a finite number m of simulator evaluations. To understand the
+behaviour of the first term, we can use the following sample complexity result for the V-statistic. The proof
+is a direct application of the triangle inequality together with Lemma 1 in Briol et al. (2019a).
+Theorem 1. Suppose that supx,x′ k(x, x′) < ∞ and let Qn consist of n iid realisations from Q ∈ Pk(X).
+Then, for any P ∈ Pk(X), we have with high probability
+|MMDk(P, Q) − MMDk(P, Qn)| = O(n− 1
+2 ).
+When �
+MMDk(Pm
+θ , Qn) is also a V-statistic approximation, both terms in (5) can be tackled with this
+result and the overall error is of size O(n− 1
+2 + m− 1
+2 ). This shows that we should take m = O(n) to ensure a
+good enough approximation of the MMD. Though this rate has the advantage of being independent of the
+dimension of X, it is relatively slow in m. We therefore require a large number of simulated data points,
+which can be computationally expensive.
+Niu et al. (2021) recently proposed an alternate approach based on randomised quasi-Monte Carlo (RQMC)
+Dick et al. (2013) samples within a V-statistic. Using stronger assumptions on U, k and Gθ, they are able to
+obtain an estimator with improved sample complexity. We now state their assumptions and result below.
+For f : X → R and a multi-index α = (α1, . . . αd) ∈ Nd, we denote the |α| = �d
+i=1 αi order partial
+derivative ∂αf = ∂|α|f/∂α1x1 . . . ∂αdxd by ∂αf. We say f ∈ Cm(X), for m ∈ N, if ∂αf exists and is
+continuous for any |α| ∈ [0, m]. For two-variable f : X × X → R, ∂α,αf is the α-partial derivative in each
+variable. The norm ∥ · ∥Lp(X) for f : X → R is defined as ∥f∥Lp(X) = (
+�
+X |f(x)|pdx)1/p. The notation
+av : b−v represents a point u ∈ [a, b]s with uj = aj for j ∈ v, and uj = bj for j /∈ v.
+Assumption A1’. The base space U = [0, 1]s, the base measure U is uniform on U, and {ui}m
+i=1 ⊂ U forms
+an RQMC point set.
+Assumption A2’. The generator Gθ : [0, 1]s → X is such that:
+1. ∂(1,...,1)Gθ,j ∈ C([0, 1]s) for all j = 1, . . . , d.
+2. for all j = 1, . . . , d and v ∈ {0, 1}s \ (0, . . . , 0), there is a pj ∈ [1, ∞], �d
+j=1 p−1
+j
+≤ 1, such that for
+g(·) = ∂vGθ,j(· : 1−v) it holds that ∥g∥Lpj ([0,1]|v|) < ∞.
+Assumption A3’. For any x ∈ X, k(x, ·) ∈ Cs(X) and ∀t ∈ Nd, |t| ≤ s, supx∈X ∂t,tk(x, x) < Ck where Ck
+is some universal constant depending only on k.
+5
+
+Theorem 2. Under A1’ to A3’ and Q ∈ Pk(X),
+|MMDk(Pθ, Q) − MMDk(Pm
+θ , Q)| = O(m−1+ϵ).
+In this case, the second term in (5) decreases at a faster rate than the first term and the overall error
+decreases as O(n− 1
+2 + m−1+ϵ) for any ϵ > 0. As a result, (ignoring log-terms) we can take m = O(n− 1
+2 ),
+meaning a much smaller number of simulations are required. However, the technical conditions required
+are either very restrictive (U must be uniform), or will be difficult to verify in practice (the conditions
+on Gθ are not very interpretable and difficult to verify). Hence, the range of cases where RQMC can be
+applied is limited. Additionally, when both k and Gθ are smooth, faster rates can be obtained using our
+optimally-weighted estimator presented in the next section.
+3. Optimally-Weighted Estimators
+We now present our estimator, which weighs simulated data. To that end, we denote the empirical measure
+of the simulated data as Pm,w
+θ
+= �m
+i=1 wiδyi where yi = Gθ(ui), and wi ∈ R is the weight associated with
+yi ∈ X for all i ∈ {1, . . . , m}. Assuming for a moment that these weights are known, then we have
+MMD2
+k(Pm,w
+θ
+, Qn) =
+m
+�
+i,j=1
+wiwjk(yi, yj) − 2
+n
+n
+�
+i=1
+m
+�
+j=1
+wjk(xi, yj) + 1
+n2
+n
+�
+i,j=1
+k(xi, xj).
+(6)
+Clearly, wi = 1/m for all i recovers the V-statistic approximation of the squared-MMD, but here we have
+additional flexibility in how to select these weights. To do so, we will make use of a tight upper bound on
+the approximation error, whose proof is in Appendix A.1.
+Theorem 3. Let c : U × U → R be a reproducing kernel such that k(x, ·) ◦ Gθ ∈ Hc and Q ∈ Pk(X). Then,
+∃K > 0 independent of {ui, yi, wi}m
+i=1 but dependent on c, k and Gθ such that:
+|MMDk(Pθ, Q) − MMDk(Pm,w
+θ
+, Q)| ≤ K × MMDc
+�
+U,
+m
+�
+i=1
+wiδui
+�
+,
+Additionally, the weights minimising this upper bound can be obtained in closed-form; i.e.
+w∗ = arg min
+w∈Rm
+MMDc
+�
+U,
+m
+�
+i=1
+wiδui
+�
+= c(U, U)−1z(U)
+(7)
+where z(U)i = µc,U(ui) =
+�
+U c(ui, u)U(du) is the kernel mean embedding of U in the RKHS Hc and
+(c(U, U))ij = c(ui, uj) for all i, j ∈ {1, . . . , m}.
+Our optimally-weighted (OW) estimator is the weighted estimator in (6) with the optimal weights in
+(7). This corresponds to estimating µk,Pθ with a weighted approximation µOW
+k,Pm
+θ
+= �n
+i=1 w∗
+i k(x, xi) =
+�n
+i=1 w∗
+i k(x, Gθ(ui)) where w∗
+i represents the importance of xi = Gθ(ui) for our approximation. To calculate
+these weights, we need to evaluate µc,U pointwise in closed-form. The key insight is that although µk,Pθ will
+usually not be available in closed-form, the same is not true for µc,U. This is because, unlike Pθ, U is usually
+a simple distribution such as a uniform, Gaussian, Gamma or Poisson. Additionally, we have full flexibility
+in our choice of c so long as k(x, ·) ◦ Gθ ∈ Hc. We refer to Table 1 in Briol et al. (2019b) or the ProbNum
+Python package Wenger et al. (2021) for a list of known closed-form kernel embeddings. Note that, both
+terms in the upper bound in Theorem 3 depend on the kernel c, meaning that c cannot simply be chosen for
+computational convenience and must also be chosen such that these quantities are as small as possible. This
+choice will be explored in further details through theory (in Section 4) and experiments (in Section 5).
+6
+
+Related methods
+The optimal weights in Theorem 3 are equivalent to Bayesian quadrature (BQ) weights
+Diaconis (1988); O’Hagan (1991); Rasmussen and Ghahramani (2002); Briol et al. (2019b). BQ is a method
+for numerical integration based on Gaussian process regression (in our case with prior mean zero and prior
+covariance function c). We can therefore think of our estimator as performing BQ to approximate all
+integrals against P in (2). This interpretation is helpful for selecting c — the kernel should be chosen so
+that the corresponding Gaussian process is a good prior for the integrands in (2). This correspondence will
+also help us derive sample complexity results in the next section.
+Our estimator minimises MMDc (U, �m
+i=1 wiδui) over the choice of weights, but we also have flexibility
+over the choice of {ui}m
+i=1. Unfortunately, this optimisation cannot be solved in closed-form, and is in fact
+usually not convex. There is a wide range of methods which have been proposed to do point selection so as
+to minimise an MMD with equally-weighted points. Kernel thinning Dwivedi and Mackey (2021), support
+points Mak and Joseph (2018) and Stein thinning Riabiz et al. (2020) are methods based on the MMD to
+subsample points given a large dataset. Kernel herding Chen et al. (2010); Bach et al. (2012) and Stein
+points Chen et al. (2018, 2019) are sequential point selection methods which use an MMD as objective. In
+addition, similar point selection methods have also been proposed for BQ Gunter et al. (2014); Briol et al.
+(2015); Belhadji et al. (2019) and these are therefore closest to our OW setting.
+4. Theoretical Guarantees
+Sample complexity
+The following theorem establishes a sample complexity of O(m− νc
+s − 1
+2 ) for our optimally-
+weighted estimator, where νc is a parameter depending on the smoothness of k and Gθ. We achieve a better
+rate than RQMC under milder conditions, as discussed below.
+Assumption A1. The base space U ⊂ Rs is bounded, open, and convex, the data space X is the entire
+Rd or is bounded, open, and convex. The base measure U has a density fU : U → [CU, C′
+U] for some CU,
+C′
+U > 0, and Pθ has a density bounded above. The point set {ui}m
+i=1 ⊂ U has a fill distance of asymptotics
+hm = O(m− 1
+s ), where hm = supu∈U mini∈[1,m] ∥u − ui∥2.
+Our assumptions on U and U are milder than those of A1’, which requires U to be uniform. The
+assumptions on X and Pθ are likely to hold for simulators in practice. We replace the requirement that
+the point set {ui}m
+i=1 is RQMC with a milder assumption on the fill distance, which quantifies how far any
+point in U can get from the set {ui}m
+i=1. The fill distance asymptotics is a standard assumption that ensures
+the coverage of U; for example, it holds for regular grids, and in expectation for independent samples. For
+further examples of point sets that guarantee small fill distance, see Wynne et al. (2021).
+Assumption A2. The generator is a map Gθ : U → X such that for some integer l > s/2, any j ∈ [1, d]
+and any multi-index α ∈ Nd of size |α| ≤ l, the partial derivative ∂αGθ,j exists and is bounded from above.
+Assumption A2 is more interpretable and easier to check than A2’ (specifically part 2) as it just requires
+knowing how many derivatives Gθ has. As stated in Niu et al. (2021), a simpler condition that implies A2’
+needs Gθ to be smooth up to the order l ≥ s, which rules out the standard choices of ν ∈ {1
+2, 3
+2, 5
+2} for large
+enough s. In contrast, we only ask that l > s/2.
+Assumption A3. k is a Matérn kernel on X of order νk such that ⌊νk + d/2⌋ > s/2, or an SE kernel, and
+c is a Matérn kernel on U of order νc ≤ min(⌊νk + d/2⌋, l).
+A3 places less restrictions on the choice of k than A3’. Although both allow for k to be the SE kernel, as
+a corollary of the Sobolev embedding theorem (Adams and Fournier, 2003, Theorem 4.12), A3’ only holds
+for a Matérn k if ⌈νk⌉ ≥ s + 1 (i.e. smooth k), while our lower bound on νk is much less restrictive. The
+conditions on c are needed to ensure k(x, ·) ◦ Gθ ∈ Hc. Note that these could be weakened using the work
+7
+
+of Kanagawa et al. (2020); Teckentrup (2020); Wynne et al. (2021), but at the expense of more restrictive
+conditions on {ui}m
+i=1 in A1.
+Theorem 4. Under A1 to A3, k(x, ·) ◦ Gθ ∈ Hc holds, and for any and Q ∈ Pk(X):
+��MMDk(Pθ, Q) − MMDk(Pm,w
+θ
+, Q)
+�� = O(m− νc
+s − 1
+2 ).
+The result shows that our method has improved sample complexity over the V-statistic for any νc and
+s. Additionally, it is better than RQMC when νc > s/2. In practice, we should pick a kernel c that is as
+smooth as possible whilst not being smoother than Gθ or k, as per A3. Hence, we should take νc to be
+smaller than l and νk, the smoothness of Gθ and k, respectively. In case the smoothness of Gθ is unknown,
+the conservative choice is to take a smaller value of νc to ensure A3 is satisfied.
+Computational Cost
+The total computational cost of our method is the sum of (i) the cost of simulating
+from the model, which is O(mCgen), where Cgen is the cost of sampling one data point, and (ii) the cost of
+estimating MMD, which is O(m2 + mn + n2) for the V-statistic and O(m3 + mn + n2) for the OW estimator.
+Our method is hence slightly more expensive when m is large. However, the cost of the simulator is often
+the computational bottleneck, sometimes taking up to tens or hundreds of CPU hours per run; see Behrens
+and Dias (2015); Kirby et al. (2022). As a result, proposing data efficient likelihood-free inference methods
+Beaumont et al. (2009); Gutmann and Corander (2016); Greenberg et al. (2019) is still an active research
+area. In cases where O(mCgen) ≫ O(m3), the OW estimator is more efficient than the V-statistic as it
+requires fewer simulations to estimate the MMD. If the simulator is not costlier than estimating the MMD
+and assuming a fixed computational budget, then the OW estimator achieves lower error than the V-statistic
+if νc/s > 1/4 and assumptions A1 to A3 hold. This result is straightforwardly derived from Theorem 4, see
+Appendix A.4 for details.
+5. Numerical Experiments
+We now illustrate the performance of our OW estimator on various benchmark simulators and on challenging
+likelihood-free inference tasks. The lengthscale of kernels k and c is set using the median heuristic Garreau
+et al. (2017), unless otherwise stated. The closed-form kernel mean embeddings used in the experiments are
+derived in Appendix A.5. Our code is available at [link removed to preserve anonymity].
+5.1. Benchmarking on popular simulators
+We begin by comparing the V-statistic with our OW estimator on a number of popular benchmark simulators
+having different dimensions for U ⊆ Rs and X ⊆ Rd. The experiments are conducted for {ui}m
+i=1 being iid
+as well as RQMC points. We fix θ for each model (see Appendix B.1 for exact values) and estimate the
+MMD2 between Pm
+θ and Pn
+θ , with k and c both being the SE kernel. We set n = 10, 000 to be large in order
+to make Pn
+θ an accurate approximation of Pθ, and m = 28 so as to facilitate comparison with RQMC, which
+requires m to be a power of 2.
+The results are reported in Table 1. For RQMC points, the errors are generally either similar for the two
+estimators (g-and-k, two moons, and Lotka-volterra models) or smaller for the OW estimator (bivariate Beta
+and MA(2)), with the OW estimator achieving lower errors in all cases barring the M/G/1 queuing model.
+This is to be expected since the M/G/1 model has a discontinuous generator, and our theory therefore
+does not hold. It is also important to note that although RQMC performs very well here even without the
+optimal weights, the simulators were chosen in order to make this comparison feasible. In many cases, U
+will not be uniform and therefore the RQMC approach will not be possible to implement and only the iid
+approach is feasible.
+8
+
+Table 1.: Average and standard deviation (in parenthesis) of estimated MMD2 (×10−3) between Pm
+θ and Pn
+θ
+computed over 100 runs for the V-statistic and our optimally-weighted (OW) estimator. Settings:
+n = 10, 000, m = 256.
+Model
+s
+d
+References
+IID V-stat
+IID OW (ours)
+RQMC V-stat
+RQMC OW (ours)
+g-and-k
+1
+1
+Bharti et al. (2022b); Niu et al. (2021)
+2.25 (1.52)
+0.086 (0.049)
+0.060 (0.037)
+0.059 (0.037)
+Two moons
+2
+2
+Lueckmann et al. (2021); Wiqvist et al. (2021)
+2.36 (1.94)
+0.057 (0.054)
+0.056 (0.044)
+0.055 (0.044)
+Bivariate Beta
+5
+2
+Nguyen et al. (2020); Niu et al. (2021)
+2.13 (1.17)
+0.555 (0.227)
+0.222 (0.111)
+0.193 (0.088)
+MA(2)
+12
+10
+Marin et al. (2011); Nguyen et al. (2020)
+2.42 (0.796)
+0.705 (0.107)
+0.381 (0.054)
+0.322 (0.052)
+M/G/1 queue
+10
+5
+Pacchiardi and Dutta (2021); Jiang (2018)
+2.52 (1.19)
+1.71 (0.568)
+0.595 (0.134)
+0.646 (0.202)
+Lotka-Volterra
+600
+2
+Briol et al. (2019a); Wiqvist et al. (2021)
+2.13 (1.10)
+2.04 (0.956)
+1.44 (0.955)
+1.42 (0.942)
+For the iid points, the improvement in performance is much more significant. The OW estimator achieves
+the lowest error for all the models when {ui}m
+i=1 are taken to be iid uniforms. Its error is reduced by a factor
+of around 20 and 40 for the g-and-k and the two moons model, respectively, compared to the V-statistic.
+As expected from our sample complexity results, the magnitude of this improvement reduces as s (the
+dimension of U) increases. However, the OW estimator still performs slightly better than the V-statistic for
+the Lotka-Volterra model where s = 600.
+5.2. Multivariate g-and-k distribution
+We now assess the impact of various practical choices on the performance of our method. To do so, we
+consider the multivariate extension of the g-and-k distribution introduced in Drovandi and Pettitt (2011)
+and used as a benchmark in Li et al. (2017b); Jiang (2018); Nguyen et al. (2020). This flexible parametric
+family of distributions does not have a closed-form likelihood, but is easy to simulate from. We define a
+distribution in this family through (Gθ, Uθ), where
+Gθ(u) = θ1+ θ2
+�
+1+0.81 − exp(−θ3z(u))
+1 + exp(−θ3z(u))
+��
+1+z(u)2�θ4z(u),
+with θ = (θ1, θ2, θ3, θ4, θ5), z(u) = Σ
+1
+2 u and U = N(0, Is), where Σ ∈ Rd×d is a symmetric tri-diagonal
+Toeplitz matrix such that Σii = 1 and Σij = θ5. The parameters θ1,θ2,θ3, and θ4 govern the location,
+scale, skewness, and kurtosis respectively, and s = d. An alternative formulation is through (˜U, ˜Gθ) where
+˜U = Unif(0, 1)s, and ˜Gθ = Gθ ◦ Φ−1 where Φ is the cumulative distribution function of a N(0, 1).
+Varying choice of k and c
+We first investigate the performance of our OW estimator for different
+combinations of k and c, the choices being either the SE or the Matérn kernel. We estimate the squared-
+MMD for each of these combinations as a function of m, with d = 10 and n = 10, 000. The Lebesgue
+measure formulation is used while computing the embeddings for both the kernels. The Matérn kernel is set
+to order νk = νc = 2.5, and the parameter value to θ0 = (3, 1, 0.1, 0.1, 0.1). The resulting curves are shown
+in Figure 2a. Our method performs best when k is the SE kernel, i.e., when it is infinitely smooth. The
+performance degrades slightly when k is Matérn, while the combination of c as SE and k as the Matérn
+kernel is the worst. This is expected from our theory, and is because the composition of Gθ and k is not
+smooth, but we approximate it with an infinitely smooth function. Hence, from a computational viewpoint,
+it is always beneficial to take k to be very smooth.
+Varying dimensions s and d
+We now analyse the impact of the choice of measure, either Gaussian or
+uniform. Figure 2b shows the OW and V-statistic estimators as the dimension s = d varies. The parameter
+values are the same as before, m = 500, and the SE kernel is used for both k and c. We observe that the OW
+9
+
+100
+200
+300
+400
+500
+No. of samples, m
+10
+3
+10
+2
+MMD2
+k(
+m,
+n)
+s = d = 10, n = 10,000
+c: SE, k: Matern
+c: Matern, k: Matern
+c: SE, k: SE
+c: Matern, k: SE
+(a)
+25
+50
+75
+100
+Dimension, s
+10
+4
+10
+3
+n = 10,000, m = 500
+V-statistic
+: Uniform
+: Gaussian
+(b)
+0.0
+0.2
+0.4
+0.6
+0.8
+4
+10
+4
+10
+3
+3 = 0.1, s = d = 10
+V-statistic
+: Uniform
+: Gaussian
+(c)
+10
+2
+10
+1
+100
+Total cost [seconds]
+10
+3
+10
+2
+n=200
+n=500
+n=1000
+n=2000
+V-statistic
+OW (ours)
+(d)
+Figure 2.: Error in estimating MMD2 for the multivariate g-and-k distribution. (a) Error of our OW estimator
+for different choices of k and c. Increasing the smoothness of k improves the performance. (b)
+Comparison of V-statistic and OW estimator as a function of dimension. OW performs better
+for both parametrisations of U, with the Gaussian giving lowest error. (c) Value of θ4 also
+impacts the performance of the OW estimator. (d) Error vs. total computation cost for different
+n.
+OW performs better than V-statistic for similar cost: m = n for V-statistic, whereas
+m = (68, 126, 200, 317) for OW.
+estimator performs better than the V-statistic even in dimensions as high as 100. In lower dimensions, the
+Gaussian embedding achieves lower error than the uniform for this model, with their performance converging
+around d = 60. This is likely due to the fact that ˜Gθ is an easier function to approximate than Gθ, but this
+is harder to assess a-priori for the user and highlights some open questions not yet covered by our theory.
+Varying model parameters
+Building on the previous result, we show that the performance of the OW
+estimation is also impacted by θ. In Figure 2c, we analyse the performance of the estimators as a function of
+parameter θ4. The SE kernel is used for both k and c. While the V-statistic is not impacted by the choice
+of θ4, the performance of our estimators degrade as θ4 increases. The behaviour is similar on varying θ3,
+albeit not as drastic as θ4, see Appendix B.2 for the plot. We expect that this difference in performance is
+due to the regularity of the generator varying with θ.
+Performance vs. computational cost
+Finally, since the OW estimator tends to be more computationally
+expensive and this simulator is relatively cheap (≈ 1 ms to generate one sample), we also compare estimators
+for a fixed computational budget. To that end, we vary n and take m = n for the V-statistic and m = 2n2/3
+for the OW estimator. Figure 2d shows their performance with respect to their total computational cost,
+including the cost of simulating from the model (d = s = 5). We see that the OW estimator achieves
+lower error on average than the V-statistic. Hence, it is preferable to use the OW estimator even for a
+computationally cheap simulator like the multivariate g-and-k.
+Composite goodness-of-fit test
+We demonstrate the performance of our method when applied to composite
+goodness-of-fit testing, using the method proposed by Key et al. (2021) with a test statistic based on the
+squared-MMD. Given iid draws from some distribution Q, the test considers whether Q is an element of
+some parametric family {Pθ : θ ∈ Θ} (null hypothesis) or not (alternative hypothesis). The approach uses a
+parametric bootstrap (Stute et al., 1993) to estimate the distribution of the squared-MMD under the null
+hypothesis, which can then be used to decide whether or not to reject. This requires repeatedly performing
+two steps: (i) estimating a parameter value through an MMD estimator of the form in Equation (3), and (ii)
+estimating the squared-MMD between Q and the model at the estimated parameter value. See Appendix B.3
+10
+
+Table 2.: Fraction of repeats for which the null was rejected. An ideal test would have 0.05 when the null
+holds, and 1 otherwise.
+Cases
+IID V-stat
+IID OW (ours)
+θ4 = 0.1 (null holds)
+0.040
+0.047
+θ4 = 0.5 (alternative holds)
+0.040
+0.413
+for the full algorithm. This needs to be done up to B times, where B can be in the hundreds or thousands,
+which can be a significant challenge computationally. This limits the number of simulated samples m that
+can be used at each step, and is therefore a prime use case for our OW estimator.
+We performed this test with a level of 0.05 using the V-statistic and OW estimator, using B = 200. We
+considered the multivariate g-and-k model with unknown θ1, θ2, θ3, and θ5 but fixed θ4 = 0.1. We used
+m = 100 and n = 500 and considered two cases: Q is a multivariate g-and-k with θ4 = 0.1 (null holds) or
+θ4 = 0.5 (alternative holds). When the null hypothesis holds, we should expect the tests to reject the null at
+a rate close 0.05, whereas when the alternative holds, we should reject at a rate close to 1. Table 2 shows
+that our test based on the OW estimator performs significantly better in that respect than the V-statistic.
+This is due to the fact that the OW estimator is able to improve both the estimate of the parameter (see
+Figure 5 in Appendix B.3), and the estimate of the test statistic, thus improving the overall performance.
+5.3. Large scale offshore wind farm model
+Finally, we consider a low-order wake model Niayifar and Porté-Agel (2016); Kirby et al. (2023) for large-scale
+offshore wind farms. The model simulates an estimate of the farm-averaged local turbine thrust coefficient
+Nishino (2016), which is an indicator of the energy produced. The parameter θ is the angle (in degrees)
+at which the wind is blowing. The turbulence intensity is assumed to have zero-mean additive Gaussian
+noise (i.e. U = N(0, 10−3)), which then goes through the non-linear mapping of the generator. Although
+this model is an approximation of the state-of-the-art models that can take around 100 CPU hours per run
+(see e.g. Kirby et al. (2022)), one realisation from this model takes ≈ 2 mins, which is still computationally
+prohibitive for likelihood-free inference. This example is indicative of the expensive simulators which are
+widely used in science, and is thus suitable for our method.
+We apply the ABC method of (4) to estimate θ with both the OW estimator and the V-statistic. The
+tolerance threshold ε is taken in terms of percentile, i.e., the proportion of the data that yields the least
+MMD distances. We use 1000 parameter values from the Unif(0, 30) prior on Θ. As the cost of the model
+far exceeds that of estimating the MMD, we take m = 10 for both estimators. With few m, setting
+the lengthscale of c using median heuristic is difficult, so we fix it to be 1. The simulated datasets took
+≈ 245 hours to generate, while estimating the MMD took around 0.13 s and 0.36 s for the V-statistic and
+the OW estimator, respectively.
+The resulting posteriors, which are approximations of the ABC posterior obtained if the MMD was
+computable in closed-form, are in Figure 3. We observe that the OW estimator’s posterior is much more
+concentrated around the true value than that of the V-statistic for both values of ε. This is because the OW
+estimator approximates the MMD more accurately than the V-statistic for the same m. Hence, our method
+can achieve similar performance as the V-statistic with much smaller m, saving hours of computation time.
+11
+
+0
+10
+20
+30
+0.00
+0.05
+0.10
+0.15
+Density
+= 5%
+V-stat.
+OW
+0
+0
+10
+20
+30
+0.00
+0.05
+0.10
+= 10%
+Figure 3.: ABC posteriors for the wind farm model. Our OW estimator yields posterior samples that are
+more concentrated around the true θ0 than the V-statistic. Settings: n = 100, θ0 = 20.
+6. Conclusion
+We proposed an optimally-weighted MMD estimator which has improved sample complexity than the
+V-statistic when the generator and kernel are smooth and the dimensionality is small or moderate. Thus, our
+estimator requires fewer data points than alternatives in this setting, making it especially advantageous for
+computationally expensive simulators which are widely used in the natural sciences, biology and engineering.
+However, a number of open questions remain, and we highlight the most relevant below.
+The parameterisation of a simulator through a generator Gθ and a measure U is usually not unique, and it
+is often unclear which parameterisation is most amenable to our method. One approach would be to choose
+a parameterisation where the dimension of U is small so as to improve the convergence rate. However, our
+result in Theorem 4 also contains rate constants which are difficult to get a handle on, and it is therefore
+difficult to identify which parameterisation is best amongst those with fixed smoothness and dimensionality.
+Finally, our sample complexity result could be extended. One limitation is that we focus on the MMD
+and not its gradient, meaning that our results are not directly applicable for gradient-based likelihood-free
+inference such as the method used for our g-and-k example Briol et al. (2019a). A future line of work
+could also investigate if our ideas translate to other distances used for likelihood-free inference, such as the
+Wasserstein distance Bernton et al. (2019) and Sinkhorn divergence Genevay et al. (2018, 2019).
+Acknowledgements
+AB was supported by the Academy of Finland (Flagship programme: Finnish Center for Artificial Intelligence
+FCAI). MN and OK acknowledge support from UKRI under the EPSRC grant number [EP/S021566/1].
+MN was also supported through The Alan Turing Institute’s Enrichment Scheme. SK was supported by
+the UKRI Turing AI World-Leading Researcher Fellowship, [EP/W002973/1]. FXB was supported by the
+Lloyd’s Register Foundation Programme on Data-Centric Engineering and The Alan Turing Institute under
+the EPSRC grant [EP/N510129/1], and through an Amazon Research Award on “Transfer Learning for
+Numerical Integration in Expensive Machine Learning Systems”.
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+Supplementary Materials
+In Appendix A, we present the proofs and derivations of all the theoretical results in our paper, while
+Appendix B contains additional details regarding our experiments.
+A. Proof of Theoretical Results
+In this section, we prove Theorems 3 and 4 and intermediate results required, and expand on the technical
+background.
+A.1. Proof of Theorem 3
+Proof. Let Pm,w
+θ
+= �m
+i=1 wiδyi = �m
+i=1 wiδGθ(ui). Using the fact that the MMD is a metric, we can use the
+reverse triangle inequality to get
+��MMDk(Pθ, Q) − MMDk(Pm,w
+θ
+, Q)
+�� ≤ MMDk(Pθ, Pm,w
+θ
+).
+To bound the right-hand side, we can use the fact that MMDk is an integral-probability metric with
+underlying function class being the unit-ball in Hk:
+MMDk(Pθ, Pm,w
+θ
+) =
+sup
+∥f∥Hk≤1
+����
+�
+X
+f(x)Pθ(dx) −
+�
+X
+f(x)Pm,w
+θ
+(dx)
+����
+=
+sup
+∥f∥Hk≤1
+�����
+�
+U
+f(Gθ(u))U(du) −
+m
+�
+i=1
+wif(Gθ(ui))
+����� .
+A similar definition for MMDc can also give us:
+�����
+�
+U
+f(Gθ(u))U(du) −
+m
+�
+i=1
+wif(Gθ(ui))
+����� ≤ ∥f ◦ Gθ∥Hc × MMDc
+�
+U,
+m
+�
+i=1
+wiδui
+�
+.
+Putting the last two results together, we get:
+��MMDk(Pθ, Q) − MMDk(Pm,w
+θ
+, Q)
+�� ≤
+sup
+∥f∥Hk≤1
+∥f ◦ Gθ∥Hc × MMDc
+�
+U,
+m
+�
+i=1
+wiδui
+�
+.
+From our assumption that k(x, ·) ◦ Gθ ∈ Hc for any x ∈ X, we have that
+K =
+sup
+∥f∥Hk≤1
+∥f ◦ Gθ∥Hc < ∞,
+and therefore
+|MMDk(Pθ, Q) − MMDk(Pm,w
+θ
+, Q)| ≤ K × MMDc
+�
+U,
+m
+�
+i=1
+wiδui
+�
+.
+To prove the second result, we note that
+arg min
+w∈Rm
+MMDc
+�
+U,
+m
+�
+i=1
+wiδui
+�
+= arg min
+w∈Rm
+MMD2
+c
+�
+U,
+m
+�
+i=1
+wiδui
+�
+.
+18
+
+and
+MMD2
+c
+�
+U,
+m
+�
+i=1
+wiδui
+�
+=
+�
+U
+�
+U
+c(u, v)U(du)U(dv) − 2
+m
+�
+i=1
+wi
+�
+U
+c(ui, u)U(du) +
+m
+�
+i,j=1
+wiwjc(ui, uj).
+The latter is a quadratic form in w, meaning it can be minimised in closed-form over w and the optimal
+weights are given by w∗. This completes the proof of the second part of the theorem.
+A.2. Chain rule in Sobolev spaces
+The proof of Theorem 4, specifically the result k(x, ·) ◦ Gθ ∈ Hc for Matérn k and c, will use a specific form
+of a chain rule for Sobolev spaces. We justify the choice of Matérn kernels—or more generally, kernels the
+RKHS of which is norm-equivalent to the well-studied Sobolev space—and prove the form of the chain rule
+for Sobolev spaces that will imply k(x, ·) ◦ Gθ ∈ Hc.
+For general c and k, k(x, ·) ◦ Gθ ∈ Hc is non-trivial to check. Here, we introduce sufficient conditions
+on c, k, and Gθ that are easily interpretable and correspond to common practical settings. Specifically,
+we consider c and k the RKHS of which, Hc and Hk, are Sobolev spaces, and Gθ of a certain degree of
+smoothness—which reduces the problem to a form of a chain rule for Sobolev spaces.1 The rest of the
+section proceeds as follows: first, we introduce the background definitions and results, then show that the
+required form of the chain rule holds for first order derivatives (Lemma 1), and finally extend the result to
+higher order derivatives (Theorem 6).
+Background.
+We consider the well-studied Sobolev kernels (see e.g. Wendland, 2005, Chapter 10), which
+are kernels that induce a reproducing kernel Hilbert space (RKHS) that is norm-equivalent to a Sobolev
+space Wl,2(X), X ⊆ Rd, for some integer l > d/2. We give the definition of Wl,2 Sobolev spaces below,
+and refer to Adams and Fournier (2003) for an in-depth treatment of Sobolev spaces and Berlinet and
+Thomas-Agnan (2004) for general RKHS theory.
+Definition 1 (Sobolev spaces). Suppose X is an open subset of Rd. The Sobolev space Wl,2(X), l > d/2, is
+a space of functions f : X → R such that ∥f∥2
+L2(X) =
+�
+X f2(x)dx < ∞, and for any multi-index α ∈ Nd with
+|α| = �d
+i=1 αi ≤ l, the weak derivative Dαf = Dα1
+x1 . . . Dαd
+xd f exists and ∥Dαf∥L2(X) < ∞.
+A weak derivative is a generalisation of the concept of a derivative to functions that are not differentiable.
+A locally integrable function Dxif is a weak derivative of f in xi if it closely resembles the behavior of
+the ordinary derivative on any open U ⊆ X: for any infinitely continuously differentiable function with a
+compact support, the integration chain rule holds with f and Dxif—as it would for an ordinary derivative.
+As the definition is only concerned with equality of the integrals in the chain rule, a weak derivative is not
+uniquely defined: two functions g1 and g2 can be weak derivatives of f in xi if (and only if) they only differ
+on a zero-volume set, meaning a set the Lebesgue measure of which is zero. As such, by Dxif we will refer
+to any function that satisfies the definition of a weak derivative. For a multi-index α = (α1, . . . αd) ∈ Nd, by
+Dαf we denote the |α| order weak derivative Dαf = Dα1
+x1 . . . Dαd
+xd f, where
+Dnxif = Dxi . . . Dxi
+�
+��
+�
+n
+f for any n ∈ N.
+1Though various forms of the chain rule for Sobolev spaces exist in the literature (for example, Evans and Garzepy (2018,
+Section 4.2.2)), they tend to either consider F ◦ f, where f is in the Sobolev space (rather than F), or place overly strong
+assumptions on f.
+19
+
+If an ordinary derivative ∂αf = ∂|α|f/∂α1x1 . . . ∂αdxd exists, it is equal to any weak Dαf. It is important
+to clarify that the definition of Sobolev spaces given here is specific to the case Wl,2(X), l > d/2. General
+Sobolev spaces W l,p(X) are subspaces of more general Lebesgue spaces, and are spaces not of functions, but
+of equivalence classes of functions. Two functions f1, f2 are in the same equivalence class [f] if they are equal
+almost everywhere. General Lebesgue and Sobolev space theory requires careful handling of the notion of
+equivalence classes, as the functions in them may differ arbitrarily on sets of Lebesgue measure zero. However,
+by Sobolev embedding theorem (Adams and Fournier, 2003, Theorem 4.12) every element of Wl,2(X) is
+continuous if l > d/2, which implies that every equivalence class contains exactly one function—and we may
+define Wl,2(X) as a space of functions, as is done above.
+Throughout the proofs, we will say f ∈ L∞(X) if it is bounded on X, and f ∈ Cm(X), for m ∈ N, if
+∂αf exists and is continuous for any |α| ∈ [0, m]. Specifically, C0(X) is the space of continuous functions,
+and C∞(X) a space of infinitely differentiable functions with continuous derivatives. The output space of
+functions in both L∞(X) and Cm(X) is omitted from the notation as it will be clear from the specific f in
+question.
+We start by recalling an important result that characterises Sobolev functions as limit points of sequences
+of C∞(X) functions. Since it is a necessary and sufficient condition, we will use this result both to operate
+on a function in a Sobolev space using the "friendlier" smooth functions, and to prove a function of interest
+lies in a Sobolev space by finding a sequence of smooth function that approximates it accordingly.
+Theorem 5 (Theorem 3.17, Adams and Fournier (2003)). For an open set X ⊆ Rd, a function f : X → R
+lies in the Sobolev space W1,2(X) and has weak derivatives Dxj[f], j ∈ [1, d] if and only if there exists a
+sequence of functions fn ∈ C∞(X) ∩ W1,2(X) such that for j ∈ [1, d]
+∥f − fn∥L2(X) → 0,
+n → ∞,
+(8)
+����Dxj[f] − ∂fn
+∂xj
+����
+L2(X)
+→ 0,
+n → ∞,
+(9)
+where ∂fn
+∂xj is the ordinary derivative of fn with respect to xj.
+Note that the functions fn converge to f in the Sobolev W1,2(X) norm, ∥f −fn∥2
+W1,2(X) = ∥f −fn∥L2(X) +
+�d
+j=1 ∥Dxjf − ∂fn/∂xj∥L2(X) → 0 as n → ∞ , if and only if (8) and (9) hold.
+Chain rule for W1,2.
+We now prove that chain rule holds for ϕ ◦ Gθ for ϕ in a Sobolev space W 1,2(X).
+For clarity, we will explicitly state the assumptions on Gθ in the main text. Recall that a measure Pθ on
+X ⊆ Rd is said to be a pushforward of a measure U on U ⊆ Rs under Gθ : U → X if for any X-measurable
+f : X → R it holds that
+�
+X f(x)Pθ(dx) =
+�
+U [f ◦ Gθ] (u)U(du).
+Lemma 1 (Chain rule for W1,2). Suppose
+• ϕ ∈ W1,2(X).
+• U ⊂ Rs is bounded, X ⊂ Rd is open, and X = Gθ(U) for some Gθ = (Gθ,1, . . . , Gθ,d)⊤. The partial
+derivative ∂Gθ,j/∂ui exists and |∂G��,j/∂ui| ≤ CG for some CG for all i ∈ [1, s] and j ∈ [1, d].
+• U is a probability distribution on U that has a density fU : U → [CU, ∞) for CU > 0.
+• Pθ is a pushforward of U under Gθ, and has a density fPθ such that fPθ(x) ≤ CPθ for all x ∈ X for
+some CPθ.
+Then ϕ ◦ Gθ ∈ W1,2(U), and for i ∈ [1, s], its weak derivative Dui[ϕ ◦ Gθ] is equal to �d
+j=1[Dxjϕ ◦ Gθ] ∂Gθ,j
+∂ui .
+20
+
+Proof. Since X is open, by Theorem 5 there is a sequence ϕn ∈ C∞(X) ∩ W1,2(X) such that
+∥ϕ − ϕn∥L2(X) → 0,
+n → ∞,
+����Dxjϕ − ∂ϕn
+∂xj
+����
+L2(X)
+→ 0,
+n → ∞,
+The proof proceeds as follows: we show that the sequence ϕn ◦ Gθ approximates ϕ ◦ Gθ, and ∂[ϕn◦Gθ]
+∂ui
+approximates the sum in the statement of the lemma, �d
+j=1[Dxjϕ ◦ Gθ] ∂Gθ,j
+∂ui , in L2(U)–norm. Then, by
+the sufficient condition in Theorem 5, ϕ ◦ Gθ lies in W1,2(U), and its weak derivative in ui is �d
+j=1[Dxjϕ ◦
+Gθ](u) ∂Gθ,j
+∂ui (u), for any i ∈ [1, s].
+Since Pθ has a density, for any X-measurable f it holds that
+�
+X
+f(x)fPθ(x)dx =
+�
+U
+[f ◦ Gθ] (u)fU(u)du.
+Together with density bounds, this gives ∥ϕ ◦ Gθ − ϕn ◦ Gθ∥L2(U) → 0 as
+�
+U
+(ϕ ◦ Gθ(u) − ϕn ◦ Gθ(u))2 du ≤ C−1
+U
+�
+U
+(ϕ ◦ Gθ(u) − ϕn ◦ Gθ(u))2 fU(u)du
+= C−1
+U
+�
+X
+(ϕ(x) − ϕn(x))2 fPθ(x)dx
+≤ C−1
+U CPθ
+�
+X
+(ϕ(x) − ϕn(x))2 dx.
+In the same fashion, ∥Dxjϕ ◦ Gθ − ∂ϕn
+∂xj ◦ Gθ∥L2(U) → 0 since
+�
+U
+�
+Dxjϕ ◦ Gθ(u) − ∂ϕn
+∂xj
+◦ Gθ(u)
+�2du ≤ C−1
+U
+�
+U
+�
+Dxjϕ ◦ Gθ(u) − ∂ϕn
+∂xj
+◦ Gθ(u)
+�2fU(u)du
+= C−1
+U
+�
+X
+�
+Dxjϕ(x) − ∂ϕn
+∂xj
+(x)
+�2fPθ(x)dx
+≤ C−1
+U CPθ
+�
+X
+�
+Dxjϕ(x) − ∂ϕn
+∂xj
+(x)
+�2dx.
+Since ϕ and Gθ are both differentiable, the ordinary chain rules applies to ϕn ◦ Gθ,
+∂[ϕn ◦ Gθ]
+∂ui
+=
+d
+�
+j=1
+�∂ϕn
+∂xj
+◦ Gθ
+� ∂Gθ,j
+∂ui
+,
+and for any i ∈ [1, s] the convergence of derivatives ∥[Dxjϕ ◦ Gθ] ∂Gθ,j
+∂ui − ∂[ϕn◦Gθ]
+∂ui
+∥L2(U) → 0 follows since
+�
+U
+�
+d
+�
+j=1
+�
+Dxjϕ ◦ Gθ
+�∂Gθ,j
+∂ui
+− ∂[ϕn ◦ Gθ]
+∂ui
+�2
+du =
+�
+U
+�
+d
+�
+j=1
+�
+Dxjϕ ◦ Gθ − ∂ϕn
+∂xj
+◦ Gθ
+�∂Gθ,j
+∂ui
+�2
+du
+≤ 2
+d
+�
+j=1
+�
+U
+��
+Dxjϕ ◦ Gθ − ∂ϕn
+∂xj
+◦ Gθ
+�∂Gθ,j
+∂ui
+�2
+du
+≤ 2C2
+G
+d
+�
+j=1
+�
+U
+�
+Dxjϕ ◦ Gθ − ∂ϕn
+∂xj
+◦ Gθ
+�2du
+where the first inequality is using the inequality (�d
+i=1 ai)2 ≤ 2 �d
+i=1 a2
+i . This completes the proof.
+21
+
+Chain rule for Wl,2.
+To extend Lemma 1 to Sobolev spaces of order higher than 1, we need the following
+version of the weak derivative product rule, for a product of a function f in W1,2 and bounded differentiable
+function g with bounded derivatives. Other versions of the product rule—for different regularity assumptions
+on g—exist in the literature (for example, Adams and Fournier (2003)); we will require this specific form.
+Lemma 2 (Product rule). Suppose X ⊆ Rd is open, f ∈ W1,2(X), g(x) is differentiable on X, and g(x) ≤ L,
+[∂g/∂xi](x) ≤ L for all x ∈ X for some constant L. Then fg ∈ W1,2(X) and for any i ∈ [1, d],
+Dxi[fg] = [Dxif]g + f
+�
+∂g/∂xi
+�
+Proof. By the criterion in Theorem 5, there is a sequence of smooth functions fn approximating f, meaning
+�
+X
+(f(x) − fn(x))2dx → 0 as n → ∞,
+�
+X
+�
+Dxif(x) − [∂fn/∂xi](x)
+�2dx → 0 as n → ∞.
+We will show that fng approximates fg with weak derivatives taking the form [Dxif]g + f[∂g/∂xi]; by the
+aforementioned criterion, it will follow that fg ∈ W1,2(X).
+First, we establish convergence of functions. As n → ∞,
+∥fg − fng∥2
+L2(X) =
+�
+X
+(f(x)g(x) − fn(x)g(x))2 dx ≤ L2
+�
+X
+(f(x) − fn(x))2 dx → 0.
+By the ordinary chain rule, ∂[fng]/∂xi = [∂fn/∂xi]g + f[∂g/∂xi]. Then, applying triangle inequality for
+norms and the fact that (a + b)2 ≤ 2a2 + 2b2 for any a, b, we get that for n → ∞,
+����
+∂fn
+∂xi
+g + fn
+∂g
+∂xi
+− [Dxif] g − f ∂g
+∂xi
+����
+2
+L2(X)
+≤ 2
+����
+∂fn
+∂xi
+g − [Dxif] g
+����
+2
+L2(X)
++ 2
+����fn
+∂g
+∂xi
+− f ∂g
+∂xi
+����
+2
+L2(X)
+≤ 2L2
+����
+∂fn
+∂xi
+− [Dxif]
+����
+2
+L2(X)
++ 2L2 ∥fn − f∥L2(X) → 0.
+This completes the proof.
+We are now ready to extend the chain rule from order 1—proven in Lemma 1—to arbitrary order l.
+Theorem 6 (Chain rule for Wl,2). Suppose
+• ϕ ∈ Wlϕ,2(X).
+• U ⊂ Rs is bounded, X ⊂ Rd is open, and X = Gθ(U) for some Gθ = (Gθ,1, . . . , Gθ,d)⊤. For some lG
+and any |α| ≤ lG, j ∈ [1, s], the derivative ∂αGθ,j exists and is in L∞(U).
+• U is a probability distribution on U that has a density fU : U → [CU, ∞) for CU > 0.
+• Pθ is a pushforward of U under Gθ with a density bounded above.
+Then ϕ◦Gθ ∈ Wl,2(U) for l = min{lϕ, lG}, and for any k ≤ l and |α0| = k, the derivative takes an α0-specific
+(κ, β, α, η)–form
+Dα0[ϕ ◦ Gθ] =
+I
+�
+i=1
+dκi
+�
+j=1
+�
+Dβijϕ ◦ Gθ
+� κi
+�
+l=1
+∂αijlGθ,ηijl,
+(10)
+where I ∈ N, and for any i ∈ [1, I], k ≥ κi ∈ N; βij ∈ Nd is a multi-index of size κi for j ∈ [1, dκi]; αijl ∈ Ns
+is of size |αijl| ≤ k, and ηijl ∈ [1, d] for l ∈ [1, κi].
+22
+
+By saying the (κ, β, α, η) form is α0-specific, we mean that the values of I, (κ, β, α, η) depend on α0, and
+may be different for α′
+0 ̸= α0; we do not index I, (κ, β, α, η) by α0 for the sake of readability.
+Before proving this result, let us point out that the (κ, β, α, η)–form introduced in the theorem can be seen
+as a form of the Faà di Bruno’s formula which generalises the chain rule to higher derivatives (Constantine
+and Savits, 1996, Theorem 1). However, since our ultimate goal is to show ϕ ◦ Gθ ∈ Wl,2(U), and the
+expression for the derivative is simply a means for proving that, an unspecified (κ, β, α, η)–form suffices. It
+is simpler to conduct a proof for general (κ, β, α, η) without using explicit Faà di Bruno forms.
+Proof of Theorem 6. Note that ϕ ◦ Gθ ∈ Wl,2(U) if and only if ϕ ◦ Gθ ∈ Wk,2(U) for k ≤ l. We use this to
+construct a proof by induction: we show the statement holds for k = 1, and that ϕ ◦ Gθ ∈ Wk,2(U) implies
+ϕ ◦ Gθ ∈ Wk+1,2(U) if k + 1 ≤ l (and the weak derivatives take a (κ, β, α, η)-form stated in Equation (10)).
+Case k = 1:
+ϕ ◦ Gθ is in W1,2(U).
+Suppose α0 = e[m] for some unit vector e[m] = (0, . . . , 0, 1, 0, . . . , 0) where the 1 is the m’th element.
+Then, as proven in Lemma 1, De[m][ϕ ◦ Gθ] = Dum[ϕ ◦ Gθ] is equal to �d
+j=1[Dxjϕ ◦ Gθ][∂Gθ,j/∂um] =
+�d
+j=1[De[j]ϕ ◦ Gθ]∂e[m]Gθ,j, so the statement holds for I = 1, κ1 = 1, β1j = e[j], α1j1 = e[m], η1j1 = j.
+Case k implies k + 1:
+If k + 1 ≤ l and ϕ ◦ Gθ is in Wk,2(U), and for every |α0| = k Equation (10) holds for
+some α0-specific (κ, β, α, η), then ϕ◦Gθ is in Wk+1,2(U), and for any |˜α0| = k +1 there is a (˜κ, ˜β, ˜α, ˜η)–form,
+|˜κ| = ˜I,
+D˜α0[ϕ ◦ Gθ] =
+˜I
+�
+i=1
+d˜κi
+�
+j=1
+�
+D
+˜βijϕ ◦ Gθ
+� ˜κi
+�
+l=1
+∂ ˜αijlGθ,˜ηijl.
+(11)
+By induction assumption, ϕ ◦ Gθ is in Wk,2(U), so it is in Wk+1,2(U) if and only if Dα0 [ϕ ◦ Gθ] is in
+W1,2(U) for any α0 of size k. The latter can be shown by studying the (κ, β, α, η)–form that Dα0[ϕ ◦ Gθ]
+takes by (10), for some α0–specific (κ, β, α, η). Since lϕ ≥ l ≥ k + 1 (the last inequality holds by the
+induction assumption), it holds that Wlϕ,2(X) ⊆ Wl,2(X) ⊆ Wk+1,2(X). Then ϕ ∈ Wk+1,2(X), and since
+|βij| = κi ≤ k by definition of βij, we have Dβijϕ ∈ W1,2(X) for all i, j. Then by Lemma 1, its composition
+with Gθ is in W1,2(U), meaning Dβijϕ ◦ Gθ ∈ W1,2(U). Consequently, Dα0 [ϕ ◦ Gθ] as per Equation (10) is a
+sum over the product of functions in W1,2(U), and bounded functions with bounded derivatives; by Lemma 2,
+such product is in W1,2(U), and it follows that Dα0 [ϕ ◦ Gθ] ∈ W1,2(U) as well.
+Finally, we show that for any fixed |˜α0| = k + 1 there are ˜I, ˜κ, ˜β, ˜α, ˜η for which (11) holds; this will
+conclude the induction step. Suppose α0 of size k, |α0| = k, is such that ˜α0 = α0 + e[m] for some α0 (that
+is unrelated to α0 in the previous part of the proof) and a unit vector e[m] (such pair of m and α0 must
+exist as |˜α0| = k + 1). For this α0, in a slight abuse of notation, we shall say that κ, β, α, η are such that
+Dα0[ϕ ◦ Gθ] takes a (κ, β, α, η) form. Then, by the sum rule for weak derivatives and the product rule
+of Lemma 2, D˜α0 [ϕ ◦ Gθ] = Dum [Dα0 [ϕ ◦ Gθ]] takes the form
+D˜α0 [ϕ ◦ Gθ] = Dum [Dα0 [ϕ ◦ Gθ]] =
+I
+�
+i=1
+dκi
+�
+j=1
+Dum
+�
+Dβijϕ ◦ Gθ
+� κi
+�
+l=1
+∂αijlGθ,ηijl
++
+I
+�
+i=1
+dκi
+�
+j=1
+�
+Dβijϕ ◦ Gθ
+�
+∂e[m]� κi
+�
+l=1
+∂αijlGθ,ηijl
+�
+.
+(12)
+23
+
+By the product rule for regular derivatives,
+∂e[m]� κi
+�
+l=1
+∂αijlGθ,ηijl
+�
+=
+κi
+�
+l0=1
+∂αijl0+e[m]Gθ,ηijl0
+�
+l∈[1,κi]
+l̸=l0
+∂αijlGθ,ηijl.
+(13)
+Since Dβijϕ ∈ W1,2(X), the statement in Lemma 1 applies to its composition with Gθ, meaning
+Dum
+�
+Dβijϕ ◦ Gθ
+�
+=
+d
+�
+j0=1
+�
+Dxj0
+�
+Dβijϕ
+�
+◦ Gθ
+� ∂Gθ,j0
+∂um
+=
+d
+�
+j0=1
+�
+Dβij+e[j0]ϕ ◦ Gθ
+� ∂Gθ,j0
+∂um
+,
+where, recall, e[j0] is a d-dimensional unit vector with 1 as the j0’th element. Substituting these into (12),
+we get
+D˜α0[ϕ ◦ Gθ] =
+I
+�
+i=1
+dκi
+�
+j=1
+d
+�
+j0=1
+�
+Dβij+e[j0]ϕ ◦ Gθ
+� ∂Gθ,j0
+∂um
+κi
+�
+l=1
+∂αijlGθ,ηijl
++
+I
+�
+i=1
+κi
+�
+l0=1
+dκi
+�
+j=1
+�
+Dβijϕ ◦ Gθ
+�
+∂αijl0+e[m]Gθ,ηijl0
+�
+l∈[1,��i]
+l̸=l0
+∂αijlGθ,ηijl
+(14)
+Now all that is left to do is find ˜I, ˜κ, ˜β, ˜α, ˜η for which this will that the (˜κ, ˜β, ˜α, ˜η)–form similar to Equa-
+tion (10). One can already see this should be possible, due to the flexibility in the definition of (˜κ, ˜β, ˜α, ˜η)–
+forms; for completeness, we give the exact values now.
+Define κ0 = 0. Take ˜I = I + �I
+i=1 κi, and
+˜κi =
+�
+κi + 1,
+i ∈ [1, I],
+κp,
+i ∈ (I + �p−1
+h=0 κh, I + �p
+h=0 κh] for p ∈ [1, I],
+˜βij =
+�
+βi⌊j/d⌋ + e[j
+mod d],
+i ∈ [1, I], j ∈ [1, 2κi+1],
+βpj,
+i ∈ (I + �p−1
+h=0 κh, I + �p
+h=0 κh], j ∈ [1, 2κp] for p ∈ [1, I],
+˜αijl =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+αi⌊j/d⌋l,
+i ∈ [1, I], j ∈ [1, 2κi+1], l ∈ [1, κi],
+e[m],
+i ∈ [1, I], j ∈ [1, 2κi+1], l = κi + 1,
+αpjl,
+i ∈ (I + �p−1
+h=0 κh, I + �p
+h=0 κh], j ∈ [1, 2κp], l ∈ [1, κp] \ {i − I − �p−1
+h=0 κh},
+αpjl + e[m],
+i ∈ (I + �p−1
+h=0 κh, I + �p
+h=0 κh], j ∈ [1, 2κp], l = i − I − �p−1
+h=0 κh for p ∈ [1, I],
+˜ηijl =
+�
+�
+�
+�
+�
+ηi⌊j/d⌋l,
+i ∈ [1, I], j ∈ [1, 2κi+1], l ∈ [1, κi],
+j
+mod d,
+i ∈ [1, I], j ∈ [1, 2κi+1], l = κi + 1,
+ηpjl,
+i ∈ (I + �p−1
+h=0 κh, I + �p
+h=0 κh], j ∈ [1, 2κp], l ∈ [1, κp] for p ∈ [1, I],
+where j mod d is the remainder of dividing j by d. Then, (14) becomes
+D˜α0[ϕ ◦ Gθ] =
+˜I
+�
+i=1
+d˜κi
+�
+j=1
+�
+D
+˜βijϕ ◦ Gθ
+� ˜κi
+�
+l=1
+∂ ˜αijlGθ,˜ηijl.
+This completes the proof of the induction step, and the theorem.
+24
+
+A.3. Proof of Theorem 4
+Before proving the main theorem, we introduce two auxilliary lemmas, Lemmas 3 and 4. The former will
+allow us to apply the chain rule of Theorem 6 to get k(x, ·) ◦ Gθ ∈ Hc, and the latter claim the asymptotic
+rate of m−νc/s−1/2. The proof of Theorem 4 will follow.
+Given A1 to A3, all that is missing to prove k(x, ·) ◦ Gθ ∈ Hc by applying Theorem 6 is the connection
+between RKHS of Matérn kernels, and Sobolev spaces. To that end, we introduce a Lemma (that is a minor
+extension to classic results, see for instance Wendland (2005, Corollary 10.48)) that links RKHS Hk of a
+Matérn kernel k of order ν with Sobolev spaces Wl,2 for l ∈ N+. In the Lemma, we briefly refer to Bessel
+potential spaces—only for their norm-equivalence both to the RKHS of Matérn kernels, and to the Sobolev
+spaces of fractional order, which themselves lie between Sobolev spaces of integer order—and to extension
+operators, that allow us to extend results on Rd to open, connected, bounded X with a Lipschitz boundary.
+Every open, bounded, and convex X has a Lipschitz boundary (Stein, 1970); for example, this includes
+the hypercube X = (0, 1)d. For a detailed overview of Bessel potential spaces, fractional Sobolev spaces,
+and extension operators, we refer to Adams and Fournier (2003); these will only appear in the proof of the
+following Lemma.
+For a β ∈ R+ ∪ {0}, we denote the ceiling operation ⌈β⌉ = min({z ∈ N | z ≥ β}), and the rounding
+operation ⌊β⌋ = max({z ∈ N | z ≤ β}).
+Lemma 3. Suppose X = Rd, or X ⊆ Rd is open, connected, and bounded with a Lipshitz boundary, and
+k is a Matérn kernel on X of order ν. Then, the RKHS Hk induced by k lies between Sobolev spaces
+W⌈ν+d/2⌉,2(X) and W⌊ν+d/2⌋,2(X), meaning
+W⌈ν+d/2⌉,2(X) ⊆ Hk ⊆ W⌊ν+d/2⌋,2(X).
+Proof. We start by proving the result for X = Rd. By Wendland (2005, Corollary 10.13), the RKHS of a
+Matérn k is norm-equivalent to the Bessel potential space Hs(X) for s = ν + d/2. The Bessel potential
+space Hs(X), by Adams and Fournier (2003, Section 7.62), is norm-equivalent to a fractional Sobolev space
+(a Sobolev-Slobodeckij space) W s,2(X), which lies between spaces of integer order, W ⌈s⌉,2(X) ⊆ W s,2(X) ⊆
+W ⌊s⌋,2(X).
+Finally, the result W⌈s⌉,2(X) ⊆ Hk ⊆ W⌊s⌋,2(X) extends to an open connected bounded X ⊂ Rd with a
+Lipshitz boundary identically to the proof of Wendland (2005, Corollary 10.48), which makes use of the
+extension operator introduced for such X by Stein (1970).
+To show the claimed asymptotic rate, we use the following straightforward corollary of Wynne et al. (2021,
+Theorem 9).
+Lemma 4 (Corollary of Theorem 9 in Wynne et al. (2021)). Suppose for any m ≥ M ∈ N+,
+• U is a measure on a convex, open, and bounded U ⊂ Rs that has a density fU : U → [0, C′
+U] for some
+C′
+U > 0.
+• {ui}m
+i=1 are such that the fill distance hm = O(m−1/s).
+• {wi}m
+i=1 are the optimal weights obtained based on the kernel cβm and measure U, parametrised by
+βm ∈ B for some parameter space B,
+• for any β ∈ B, cβ is a Matérn kernel of order νc; νc is independent of β.
+Then, for some C0 independent of m and f, and any f ∈ Hc with ∥f∥Hc = 1,
+�����
+�
+U
+f(u)U(du) −
+m
+�
+i=1
+wif(ui)
+����� ≤ C0m−νc/s−1/2.
+25
+
+Proof. The expression on the left hand side of Wynne et al. (2021, Theorem 9) is |
+�
+U f(u)U(du) −
+�m
+i=1 wif(ui)|; the notation from their paper to this result maps as θ → β, p → fU, X → U, x → u,
+Θ → B, and the prior mean µ(β) = 0 for any β ∈ B. First, we show the assumptions in the Theorem hold.
+Assumption 1 (Assumptions on the Domain): An open, bounded, and convex U satisfies the assumption,
+as discussed in Wynne et al. (2021).
+Assumption 2 (Assumptions on the Kernel Parameters): Since cβ is a Matérn kernel of order νc, the
+smoothness constant of cβ is νc +s/2 regardless of the value of β ∈ B, meaning τ(β) = τ −
+c = τ +
+c = νc +s/2 >
+s/2. Lastly, the norm equivalence constants of Wynne et al. (2021, Equation 3) are the same for all β—since
+the respective RKHS and Sobolev spaces are—so the set of extreme values B∗
+m is finite and does not depend
+on m; we denote B∗
+c = B∗
+m, to highlight that B∗
+c only depends on the choice of kernel family c and not m.
+Assumption 3 (Assumptions on the Kernel Smoothness Range): As discussed in Assumption 2, τ(β) =
+νc + s/2 for any β ∈ B, so the set in the statement of Assumption 3 has only one element.
+Assumption 4 (Assumptions on the Target Function and Mean Function): The target function f is in Hc,
+meaning τf = τ −
+c = τ +
+c = νc + s/2. The mean function µ(β) was taken to be zero, so has zero norm.
+Lastly, take h0 such that h1 ≤ h0; as we assumed hm = O(m−1/s), it holds that h0 ≤ hm for all m ≥ 1.
+Therefore, all the assumptions are satisfied and Wynne et al. (2021, Theorem 9) applies; moreover, the
+bounding expression is C0m−α/s for α = νc + s/2 and some C0 independent of m and f since
+• hm = O(m−1/s), and as τf = τ −
+c = τ +
+c = νc + s/2 as discussed in the verification of assumptions,
+hmax(τf,τ −
+c )
+m
+= O(m−νc/s−1/2),
+• the rest of the multipliers do not depend on m and f: C depends only on U, s, τf = νc + s/2, and
+B∗; ∥fU∥L2(U) is a constant and finite since fU is bounded above; τf − τ +
+c = 0 so rising to its power
+produces 1; the norm ∥f∥Hc = 1; for any m ≥ M, µ(βm) = 0.
+This completes the proof.
+Now we are ready to prove the main theorem.
+Proof of Theorem 4. To show k(x, ·) ◦ Gθ ∈ Hc for all x ∈ X, first note that Lemma 3 applies for both
+U and X that satisfy A1: trivially for Rd, and for an open, convex, and bounded space since it has a
+Lipschitz boundary (Stein, 1970). Since by A3, k is a Matérn kernel of order νk, it holds by Lemma 3 that
+k(x, ·) ∈ Wlϕ,2(X) for lϕ = ⌊νk + d/2⌋ and any x ∈ X. Then, by Theorem 6, k(x, ·) ◦ Gθ ∈ W˜l,2(U), for a
+Gθ that satisfies A2, and ˜l = min(lϕ, l) = min(⌊νk + d/2⌋, l). By A3, νc ≤ min(⌊νk + d/2⌋, l) − s/2 = ˜l − s/2,
+and it holds that ˜l ≥ νc + s/2. Since ˜l is an integer, this implies ˜l ≥ ⌈νc + s/2⌉, and we have that
+W˜l,2(U) ⊆ W⌈νc+s/2⌉,2(U).
+Finally, as c is a Matérn kernel or order νc, by Lemma 3 it holds that
+W⌈νc+s/2⌉,2(U) ⊆ Hc, and we arrive at k(x, ·) ◦ Gθ ∈ Hc.
+Since k(x, ·) ◦ Gθ ∈ Hc holds, we can use Theorem 3 and state
+|MMDk(Pθ, Qn) − MMDk(Pm
+θ , Qn)| ≤ K × MMDc
+�
+U,
+m
+�
+i=1
+wiδui
+�
+.
+By the reproducing property, it holds that sup
+f∈Hc
+∥f∥Hc=1
+|
+�
+U f(u)P(du)−
+�
+U f(u)Q(du)| is equal to MMDc(P, Q)
+for any two distributions P, Q on U. Then,
+MMDc(U,
+m
+�
+i=1
+wiδui) =
+sup
+f∈Hc
+∥f∥Hc=1
+�����
+�
+U
+f(u)U(du) −
+m
+�
+i=1
+wif(ui)
+����� .
+The expression under the supremum is bounded by Lemma 4 with C0m−νc/s−1/2, for C0 independent of m
+and f. Therefore, MMDc(U, �m
+i=1 wiδui) ≤ C0m−νc/s−1/2, and the result holds.
+26
+
+Table 3.: Computational and sample complexity rates of the V-statistic and the OW estimator with respect
+to m.
+Cost
+Error
+V-statistic
+O(m2)
+O(m− 1
+2 )
+OW
+O(m3)
+O(m− νc
+s − 1
+2 )
+Note that while the result was formulated for the special case of convex spaces, it applies more generally to
+any open, connected, bounded X ⊂ Rd, U ⊂ Rs with Lipschitz-continuous boundaries—with no changes to
+the proof. The applicability to X = Rd remains unchanged; U, however, must remain bounded for Theorem 6
+to hold.
+A.4. Computational and sample complexity
+We derive the condition under which the OW estimator achieves better sample complexity than the V-statistic
+for the same order of computational cost, see Table 3 for the rates.
+Suppose the cost for both V-statistic and OW is O( ˜m). Then, the sample complexity for the V-statistic
+can be written in terms of ˜m as O( ˜m−1/4). Similarly, for the OW estimator, the sample complexity in terms
+of ˜m is O( ˜m−(νc+ s
+2 )/3s). The more accurate estimator is therefore the one whose error rate goes to zero
+quicker. Therefore, the OW estimator is more accurate than the V-statistic if
+νc
+s > 1
+4,
+which for the common choice of νc = 5/2 implies s < 10.
+A.5. Derivation of closed-form kernel embeddings
+We have zi =
+�
+U c(ui, v)U(dv), where c : U × U → R is the SE kernel parameterised by the lengthscale
+l > 0, i.e., c(u, v) =
+√
+2πlϕ(u; v, l2), where ϕ is the Gaussian pdf. For s > 1, we can write the kernel as
+c(u, v) = �s
+j=1 c(uj, vj). We now derive closed-form kernel embeddings for zi for different choices of the
+base space U and the distribution U.
+For U = [0, 1]s, and U the uniform distribution, i.e., ui ∼ Uniform([0, 1]s), we get
+zi =
+s
+�
+j=1
+�
+[0,1]
+c(uij, vj)dvj =
+s
+�
+j=1
+√
+2πl
+�
+ϕ(1; uij, l2) − ϕ(0; uij, l2)
+�
+,
+where ϕ is the Gaussian cdf and uij is the jth element of ui.
+In the case of U = Rs, and U being the Gaussian distribution such that ui ∼ N(µ, Σ), where µ =
+[µ1, . . . , µs]⊤ and Σ is the s-dimensional diagonal matrix with entries (σ2
+1, . . . , σ2
+s), the closed-form embedding
+for zi reads
+zi =
+s
+�
+j=1
+� ∞
+−∞
+c(uij, vj)ϕ(vj; µj, σ2
+j )dvj =
+s
+�
+j=1
+√
+2πl
+� ∞
+−∞
+ϕ(vj; uij, l2)ϕ(vj; µj, σ2
+j )dvj
+=
+s
+�
+j=1
+�
+l2
+(l2 + σ2
+j ) exp
+�
+−(uij − µj)2
+2(l2 + σ2
+j )
+�
+.
+27
+
+0.0
+0.2
+0.4
+0.6
+0.8
+3
+10
+4
+10
+3
+MMD2(
+m,
+n)
+4 = 0.1, s = d = 10
+V-statistic
+: Uniform
+: Gaussian
+0
+20
+x
+0.0
+0.1
+0.2
+0.3
+Density
+4 = 0.1
+4 = 0.5
+Figure 4.: Additional results for the multivariate g-and-k distributions. Left: Estimated MMD2 for the
+V-statistic and our OW estimator as a function of θ3. Right: Histogram of samples from the
+g-and-k distribution for different values of θ4. Settings: no. of samples = 100,000, θ1 = 3, θ2 = 1,
+θ3 = 0.1.
+For the special case of Σ = diag(σ2, . . . , σ2), the expression simplifies to
+zi =
+�
+l2
+l2 + σ2
+�s/2
+exp
+�
+− ∥ui − µ∥2
+2(l2 + σ2)
+�
+.
+B. Additional Experimental details
+True parameter values of the benchmark simulators in Section 5.1 is given in Appendix B.1. Appendix B.2
+and Appendix B.3 provide additional results and details regarding the experiments in Section 5.2. Finally,
+the link to the source code of the wind farm simulator is in Appendix B.4.
+B.1. Benchmark Simulators
+We now provide further details on the benchmark simulators. For drawing iid or RQMC points, we use
+the implementation from SciPy Virtanen et al. (2020). Below, we report the parameter value θ used to
+generate the results in Table 1 for each model. We refer the reader to the respective reference in Table 1 for
+a description of the model and their parameters.
+g-and-k distribution: (A, B, g, k) = (3, 1, 0.1, 0.1)
+Two moons: (θ1, θ2) = (0, 0)
+Bivariate Beta: (θ1, θ2, θ3, θ4, θ5) = (1, 1, 1, 1, 1)
+Moving average (MA) 2: (θ1, θ2) = (0.6, 0.2)
+M/G/1 queue: (θ1, θ2, θ3) = (1, 5, 0.2)
+Lotka-Volterra: (θ11, θ12, θ13) = (5, 0.025, 6)
+B.2. Multivariate g-and-k
+The performance of the V-statistic and our OW estimator as a function of θ3 parameter of the multivariate
+g-and-k distribution is shown in Figure 4 (left). The observed effect on the performance is similar to that of
+Figure 2c, where the error in the OW estimator increases as we vary θ3. The degradation in performance is
+not as severe as when varying θ4, indicating that the smoothness of the multivariate g-and-k generator is not
+impacted by θ3 compared to θ4. Both the uniform and the Gaussian embedding achieves better performance
+than the V-statistic, whose performance remains unaffected by θ3.
+28
+
+B.3. Composite goodness-of-fit test: details and additional results
+Algorithm 1: Composite goodness-of-fit test
+Input: Pθ, Qn, α, B
+ˆθn = arg min
+θ
+MMD2(Pθ, Qn) ;
+for k ∈ {1, . . . , B} do
+Qn
+(k) = 1
+n
+�n
+i=1 δx(k)
+i ,
+�
+x(k)
+i
+�n
+i=1 ∼ Pˆθn;
+ˆθn
+(k) = arg min
+θ∈Θ
+MMD2(Pθ, Qn
+(k));
+∆(k) = MMD2(Pˆθn
+(k), Qn
+(k));
+cα = quantile({∆(1), . . . , ∆(B)}, 1 − α);
+Pm
+ˆθn = 1
+m
+�m
+i=1 δyi, where {yi}m
+i=1 ∼ Pˆθn;
+if MMD2(Pm
+ˆθn, Qn) > cα then
+return reject;
+else
+return do not reject;
+Algorithm 2: Random-restart optimiser
+Input: Pθ, Qn, m, I, R, S, s, Θinit
+Function loss(θ) is
+Pm
+θ = 1
+m
+�m
+i=1 δyi, where {yi}m
+i=1 ∼ Pθ ;
+return MMD2(Pm
+θ , Qn);
+θtrial
+(1) , . . . , θtrial
+(I) ∼ Θinit ;
+Select θinit
+(1) , . . . , θinit
+(R) ∈ {θtrial
+(k) }I
+k=1 that yield the
+smallest loss(θinit
+(k) );
+ˆθopt
+(1) , . . . , ˆθopt
+(R) = for k ∈ {1, . . . , R} do
+ˆθopt
+(k) = adam_optimizer(loss, S, s, θinit
+(k) )
+return θ∗ ∈ {ˆθopt
+(k) }R
+k=1 s.t.
+∀k. loss(θ∗) ≤ loss(ˆθopt
+(k) );
+Algorithm 1 shows the details of the composite goodness-of-fit test using the parametric bootstrap. The
+algorithm is written for the V-statistic estimator, but each instance of the squared MMD can be replaced
+with our OW estimator. In practice, to compute arg minθ MMD2(Pθ, Qn) we use gradient-based optimisation,
+as described in Algorithm 2. The definitions of the hyperparameters of these two algorithms, and the values
+that we use, are given below:
+hyperparameter
+value
+α
+0.05
+level of the test
+B
+200
+number of bootstrap samples
+m
+100
+number of samples from the simulator
+n
+500
+number of observations in the data
+I
+50
+number of initial parameters sampled
+R
+10
+number of initial parameters to optimise
+S
+200
+number of gradient steps
+s
+0.04
+step size
+Θinit is the distribution from which the initial parameters are sampled, and is a uniform distribution with
+the following ranges: θ1 : (0.001, 5), θ2 : (0.001, 5), θ3 : (0.001, 1), θ5 : (0.001, 1). To compute the fraction of
+times that the null hypothesis is rejected (Table 2) we repeat the experiment 150 times.
+To demonstrate why the test using the OW estimator has better performance, we examine the distribution
+of the parameter estimates. We repeatedly sample Qn from the multivariate g-and-k distribution with
+θ = θ0, and then plot a histogram of ˆθn = arg minθ MMD2(Pθ, Qn). Figure 5 shows that the estimates of
+the parameters computed using the OW estimator are more concentrated around the true parameter value,
+whereas the estimates computed using the V-statistic have higher variance. This means that, when using
+the V-statistic, the distribution of the test statistic approximated by the bootstrap has higher variance, thus
+the estimated critical value is more conservative, and the test is not sensitive to smaller departures from the
+null hypothesis. In contrast, when using the OW estimator, the estimated critical value is less conservative
+and the test has higher performance.
+29
+
+2.6
+3.1
+θ1
+↔ 2
+0.9
+1.1
+θ2
+↔ 3
+−0.2
+0.5
+θ3
+↔ 10
+−0.05
+0.25
+θ5
+↔ 2
+Figure 5.: Histogram of parameters estimated using Algorithm 2 with the V-statistic estimator (blue) and
+the OW estimator (orange). The histogram is generated by sampling 100 sets of observations
+of size n from Q, and applying the minimum distance estimator to each one. The black vertical
+lines show the true value of the parameter from which the observed data was generated. The
+estimates produced using the V-statistic have high variance, thus to improve the readability of
+the plot some of the outliers are not shown as they are outside of the x-axis range of the plot. ↔
+indicates the number of these estimates for each parameter. For the OW estimator, all estimates
+are within the x-axis range. The hyperparameters were set as in the table above.
+B.4. Large scale wind farm model
+The low-order wake model is described in Kirby et al. (2023) and the code is available at https://github.
+com/AndrewKirby2/ctstar_statistical_model/blob/main/low_order_wake_model.py.
+30
+