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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 +∗ t.scheel@fz-juelich.de +† m.sega@ucl.ac.at +‡ j.harting@fz-juelich.de +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. +[1] Z. Hu and R. C. Srivastava, Evolution of raindrop size distribution by coalescence, breakup, and evaporation: Theory and +observations, J. Atmospheric Sci. 52, 1761 (1995). +[2] A. 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Fluid Mech. 872, 327 (2019). + diff --git a/0tE5T4oBgHgl3EQfOg6n/content/tmp_files/load_file.txt b/0tE5T4oBgHgl3EQfOg6n/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b6564fc73b6ba6004d2c6d1d76136b76c8e2c3cb --- /dev/null +++ b/0tE5T4oBgHgl3EQfOg6n/content/tmp_files/load_file.txt @@ -0,0 +1,698 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf,len=697 +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 1, D-91058 Erlangen, Germany 2Department of Physics, Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg, Cauerstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 1, D-91058 Erlangen, Germany 5Department of Chemical and Biological Engineering and Department of Physics, Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg, Cauerstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 1, D-91058 Erlangen, Germany (Dated: January 16, 2023) Liquid lens coalescence is an important mechanism involved in many industrial and scientific ap- plications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' The theoretical analysis of the coalescence of liquid lenses, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' This linear dependence marks the so-called viscous regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' On the other hand, analytical approaches rely on assumptions such as a reduced dimensionality (thin-sheet equation) or infinite bridge growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' In order to describe the growth dynamics of top-down symmetric liquid lenses (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 1) it has to be taken into account that they involve two principal radii of curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' As a result, the bridge is characterized not only by its height h0, but also by its width w0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' Heuristically, one can imagine the evolution of the bridge width w0 in the same ∗ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='scheel@fz-juelich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='de † m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='sega@ucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='at ‡ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='harting@fz-juelich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='de arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='05498v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='flu-dyn] 13 Jan 2023 2 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' Lengths and times are provided in lattice Boltzmann units ∆x and ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' terms as that of its height h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' In reality, however, a complex coupling between the two directions is to be expected, which is difficult to model with analytical approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' In this article, we investigate the coalescence dynamics of liquid lenses using the color gradient lattice Boltzmann simulation method [41–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' The final section provides conclusions and a short outlook on future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=', 19 specifies the lattice direction and k ∈ {1, 2, 3} the fluid component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' (5) Here, wi are the lattice weights wi = � � � 1/3 i = 1 1/18 i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' , 7 1/36 i = 8, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' , 19 (6) and φk i is the angle between the color gradient ⃗F k and the lattice direction ⃗ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' , 7 1/27 i = 8, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' , 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 4 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' LIQUID LENS COALESCENCE Our study focuses on the coalescence of two identical, top-down symmetric liquid lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' We begin our investiga- tion with the quasi two-dimensional case (cylindrical symmetry), before later turning to the fully three-dimensional simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' The droplets are initialized side by side and connected via a contact point (resolved by approximately 5 lattice nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' These quantities determine a characteristic velocity scale, also known as capillary velocity, given by the ratio vc = σ/µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' The Reynolds number of the coalescing droplets can thus be expressed as Re = ρ σ h0/µ2 [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' In this regime Re ≪ 1 and the flow is described by the Stokes equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' The crossover between the viscous and inertial regime occurs at Re ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' From then on, viscous dissipation becomes increasingly negligible and the dynamics of the system is determined by inertial forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' (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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' The color gradient lattice Boltzmann method, on the contrary, is stable over a much wider range of surface tension values [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' The droplets are initialized with a radius of 282 lattice nodes and a contact angle θ = 30◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='106 and Kv = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='21 as previously obtained from similarity solutions of the thin-sheet equation [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' All simulations show very good agreement with the analytical solution of the thin-sheet equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 5 10 3 10 1 101 103 105 107 t/tc 10 3 10 1 101 103 105 h0/hc / =11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='6 / =5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='8 / =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='696 / =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='348 / =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='16 / =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='9 / =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='0348 / =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='0116 / =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='00116 / =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='000116 h0/hc = t/tc h0/hc = (t/tc)2/3 theory Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' Power law relation for the bridge growth in 2d covering the viscous as well as inertial regime (solid line: interpolation according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' (16), dashed line: viscous theory, dotted-dashed line: inertial theory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 0 250 500 750 1000 1250 1500 1750 2000 y [Δx] 0 200 400 600 z [Δx] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='0 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' Flow field of viscous (σ/µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='000116, left) and inertial (σ/µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='348, right) liquid lens coalescence, where the grey scale of the velocity vectors represents the magnitude of the velocity vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' The velocity field in the viscous regime is inherently dipolar and approaches a plug flow inside the liquid lens phase over time – see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 3 (left panel) for a representative velocity field obtained from the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' While in the vicinity of the bridge minimum the flow field of the inertial regime is still dipolar (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 3, right panel), two additional dipolar flow structures arise approximately at the center of each of the two initial liquid lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' In analogy to the assumptions of the thin sheet equation, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 4 shows the profile uy(y, t) of the y-component of the velocity, averaged over the droplet extension along the z axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' At larger distances to h0 (|ξ| > 1), however, the simulated velocity profile starts deviating from the thin-sheet solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' Thus, viscous damping in the surrounding fluids influences the velocity field inside the droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' Next, we extend our simulations to the fully three-dimensional case (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 5), where we use a system size of 768 × 2096 × 768 lattice nodes in x, y and z direction with periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' The update of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='2 · 109 lattice sites requires a considerable amount of computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' Therefore, the simulations were conducted on the JURECA Booster machine with 32, 768 Intel KNL cores using up to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='4 million core-hours to generate a single data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' In analogy to the pseudo two-dimensional case, we initialize two equilibrated lenses with a contact angle of θ = 30◦ (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 1) and adequate spacing to the domain boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' The growth of the bridge width reported in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' This indicates that the thin-sheet equation for the 2d case fails to describe the dynamics of the three-dimensional bridge growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' The scaling law is however not in 6 15 10 5 0 5 10 15 = y h0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='75 U = uy /kv t/tc=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='6 ⋅ 10-3 t/tc=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='2 ⋅ 10-3 t/tc=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='8 ⋅ 10-3 t/tc=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='4 ⋅ 10-3 theory 10 5 0 5 10 = y h0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='0 U = uy 3 t1/3/(2ki) t/tc=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='5 ⋅ 102 t/tc=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='3 ⋅ 103 t/tc=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='0 ⋅ 103 t/tc=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='6 ⋅ 103 theory Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' Snapshot of two coalescing liquid lenses in 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' The snapshot is taken at t/tc = 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='8 (12,000 ∆t), where the connecting bridge has already developed for a capillary velocity σ/µ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='9 (inertial regime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' contradiction to the experimental data shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' [32], where reasonably the transition region between the viscous and the inertial regime was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' In the three-dimensional case the naive assumption of a decoupled width and height growth is clearly not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' The dependence is clearly of the type hc ∼ µ/σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' Our simulation method - the color-gradient lattice Boltzmann method - has proven to deliver hydrodynamically consistent results for the required wide parameter ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' This allows us to investigate the coalescence dynamics from the viscous to the inertial regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' For the pseudo two-dimensional case we find good agreement with the similarity solutions of the thin-sheet equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' In the viscous regime the bridge grows linearly with time and in the inertial regime, the bridge growth is proportional to t2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 7 10 3 10 1 101 103 105 107 t/tc 10 3 10 1 101 103 105 w0/wc / =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='9 / =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='16 / =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='696 / =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='348 / =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='0348 / =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='0116 / =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='9 / =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='16 / =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='696 / =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='348 / =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='0348 / =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='0116 / =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='00116 h0/hc = t/tc h0/hc = (t/tc)1/2 h0/hc = ( 1 t/tc + 1 (t/tc)1/2 ) 1 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' Power law relation for the bridge growth in 3d covering the viscous as well as inertial limit (solid line: interpolation according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' (16), dashed line: t, dotted-dashed line: t1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' Left panel: bridge width w0(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' right panel: bridge height h0(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 10 3 10 2 10 1 100 / 10 1 100 101 102 hc hfit c ( , ) hc ( / ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content='00 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' Dependence of the best-fit hc on the capillary velocity in 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' The dashed line represents the linear relation obtained by fitting the exponent of capillary velocity (σ/µ) to the data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' The three-dimensional coalescence simulations, on the contrary, deviate from the similarity solution of the thin- sheet equation exhibiting a t1/2 dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' This makes the dynamics of the bridge width the dominant process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' These results underline the necessity of a more generic theoretical framework for a more accurate understanding of the general coalescence process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' ACKNOWLEDGMENTS We acknowledge Jacco Snoeijer and Michiel Hack for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} +page_content=' 872, 327 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE5T4oBgHgl3EQfOg6n/content/2301.05498v1.pdf'} diff --git a/19AyT4oBgHgl3EQfPvYS/content/tmp_files/2301.00028v1.pdf.txt b/19AyT4oBgHgl3EQfPvYS/content/tmp_files/2301.00028v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1d4a35c7ee70b40881387251eea4be0d88be610c --- /dev/null +++ b/19AyT4oBgHgl3EQfPvYS/content/tmp_files/2301.00028v1.pdf.txt @@ -0,0 +1,1463 @@ +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: aurore.loisy@bristol.ac.uk +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. +REFERENCES +Aboulhasanzadeh, B. & Tryggvason, G. 2014 Effect of bubble interactions on mass transfer +in bubbly flow. 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Proceedings of the London Mathematical +Society s2-20 (1), 196–212. + diff --git a/19AyT4oBgHgl3EQfPvYS/content/tmp_files/load_file.txt b/19AyT4oBgHgl3EQfPvYS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fc4fdb9a3301578139ee077b1a2c599bf4dd5cfe --- /dev/null +++ b/19AyT4oBgHgl3EQfPvYS/content/tmp_files/load_file.txt @@ -0,0 +1,815 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf,len=814 +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' revised xx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' We first extend the theory of Koch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Scalar transport in arrays of freely rising bubbles is considered by us subsequently, using numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' In this case, the dispersion is found to be convectively enhanced at low P´eclet number, like in ordered arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Introduction Bubble columns are commonly used in a broad range of technologies, notably in the chemical and biochemical industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Simple bubble columns do not require active stirring and can therefore operate without interior moving parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The large surface area between gas and liquid is useful for mass and species transfer, possibly involving chemical reactions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=', Deckwer 1992) such as in air-lift bioreactors, an example of which is in the treatment of wastewater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Bubble columns are also used for this reason in direct contact heat transfer (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=', Hewitt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=', Batchelor 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Koch & Brady 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=', Alm´eras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' ‡ Email address for correspondence: aurore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='loisy@bristol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='uk arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='00028v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='flu-dyn] 30 Dec 2022 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Loisy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Naso, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Spelt a chemical species, or the temperature) through incompressible bubbly flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' If these effects are small, as assumed herein, temperature and solute concentration can be considered as passive scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Koch & Brady 1985, 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' This effective diffusivity is defined from an Eulerian perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Experimentally, scalar dispersion is usually investigated from a Lagrangian point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=', the positions, shapes, and orientations of the inclusions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' This particular situation is essentially relevant to heat and electricity conduction in composite materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' When Pe ≫ 1, the dominant contribution to the effective diffusivity is due to convective mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' This last regime is that generally encountered in bubbly flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Recently Alm´eras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (2015) investigated experimentally the dispersion of a low- diffusive dye within a homogeneous swarm of high-Reynolds-number rising bubbles at Pe = O(106);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' herein we define the Reynolds number as Re = Udb/νc where νc is the kinematic viscosity of the liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' They showed that scalar mixing primarily results from pseudo-turbulence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=', 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Apart from the work of Alm´eras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' To the best of our knowledge, neither theoretical nor numerical investigations of scalar mixing in homogeneous bubbly flows have been reported thus far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Theoretical work is, however, available for other types of multiphase systems, and we shall review these now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The determination of such an effective diffusivity, at the macroscale, necessitates consideration of the conditions at the microscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=', Maxwell (1873), Jeffrey (1973)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Another class of analytical studies assumes a periodic microstructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Sangani & Acrivos 1983), and the effect of anisotropy has been investigated by considering periodic arrangements of spheroidal inclusions (Kushch 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Harfield 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Koch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' For these reasons, prior analyses are, a priori, not applicable to bubbly suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Nevertheless, we showed in prior work (Loisy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' In view of the already significant number of parameters involved, we shall therefore adopt this restriction here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The theoretical framework and problem statement are provided in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The first objective (in section 4) is to 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Loisy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Naso, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Spelt elucidate the role played by liquid inertia in ordered suspensions, using direct numerical simulation and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Finally, the main results and perspectives of this work are provided in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Problem statement The local evolution of the passive scalar c in each fluid is governed by ∂c ∂t + ∇ · q = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='1a) where q is the flux of scalar given by q = uc − D∇c (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='1b) with u the fluid velocity and D the constant scalar diffusivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' We assume that the scalar and its gradient are continuous across the interface, and phase change is not considered in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Under these assumptions, no distinction between the phases is needed for the scalar transport, which is described by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='1) in the entire system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' We return to these restrictions in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='1 and in the Conclusions section;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' the objective here is to study this key basic reference problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' In the context of heat transfer, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' In the context of mass transfer, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=', carbon dioxide dispersion in the air-water system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=', fluid properties, gas vol- ume fraction) but different microscopic configurations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=', bubble individual positions, shapes and velocities), and average over those realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' In concrete terms, ensemble averaging would be realized by averaging over a large number of experiments run under identical macroscopic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The ensemble-averaged transport equation is obtained from ensemble averaging the local transport equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' It reads ∂⟨c⟩ ∂t + ∇ · ⟨q⟩ = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='3) where the velocity fluctuations are defined by u′ = u − ⟨u⟩ and the scalar fluctuations by c′ = c − ⟨c⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (ii) −D∇⟨c⟩ is the diffusion of the average scalar field directly by the average scalar gradient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (iii) ⟨u′c′⟩ corresponds to the advection of the scalar fluctuations by the velocity fluctuations induced by bubble motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' When the suspension is statistically homogeneous and in a statistically stationary state, the linearity in c of the local flux (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='4) where Deff is a constant effective diffusivity tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Comparison of the effective diffusivity definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='4) with the average flux expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='3) yields the expression of the effective diffusivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='5) where Dconv · ∇⟨c⟩ = −⟨u′c′⟩ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='6) is the convective contribution arising from bubble-induced velocity fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' We recall here that further contributions to the average flux (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='3) and hence to the effective diffusivity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='5) arise if the diffusivity in the fluids are not the same, or if the concentration is discontinuous at fluid/fluid interfaces (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=', Batchelor & O’Brien (1977);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Koch & Brady (1985)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' We return to the significance of this in section 6 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' In the limit Nb = 1, one obtains a simple cubic array of bubbles, which is of interest as a model of perfectly ordered suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The bubbles rise under the sole effect of buoyancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='7) 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Loisy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Naso, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Our main objective is to elucidate the effect of the value of Pe on the effective diffusivity using analytical and numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The effective diffusivity necessarily also depends on the gas volume fraction φ = (Nbπd3 b)/(6h3) (h is the linear size of the unit cell);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The ranges of φ, Nb and Re studied here are summarized in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' As the bubbly flows we consider are buoyancy-driven,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' a difficulty arises from the fact that U is a priori unknown,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' and depends in a complex manner on Nb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' the density and viscosity ratios between both phases,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' the Archimedes (or Galileo) number Ar = � ρc|ρd − ρc|gd3 b/µc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' and the Bond (or E¨otv¨os) number Bo = |ρd − ρc|gd2 b/γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' where the subscripts d and c refer to the disperse (gas) and continuous (liquid) phases,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' g is the magnitude of the gravitational acceleration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' ρ denotes density,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' µ is the dynamic viscosity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' and γ is the surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' In most bubbly flows of practical relevance, the gas-to-liquid density and viscosity ratios are vanishingly small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Their precise values are not important from a physical point of view as long as they are small enough;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' in the simulations, the gas-to-liquid density and viscosity ratios were set to ρd/ρc = 10−3 and µd/µc = 10−2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The dependence of U on (Ar, Bo, φ, Nb) has been addressed in Loisy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (2017) and is not further discussed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' In the present study, we shall therefore assume that U is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The effective diffusivity of ordered and freely evolving bubbly suspensions 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Methodology For convenience of numerical implementation, we reorganise the problem formulation by introducing the decomposition c = ¯c + ˜c (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='1) where ¯c is the imposed constant linear scalar field ¯c = ∇⟨c⟩ · x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='2) The advantage of this decomposition is that the disturbance field ˜c is then spatially periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The governing equation for this disturbance field is ∂˜c ∂t + ∇ · (u˜c) − ∇ · (D∇˜c) = −u · ∇⟨c⟩ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='3) which is the equation we integrate numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The convective contribution to the effective diffusivity is then calculated from Dconv · ∇⟨c⟩ = −⟨u′˜c⟩ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='4) which can be shown to be equivalent to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' In this expression, ⟨ ⟩ is defined as an ensemble average operator, as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' For statistically homogeneous and stationary systems, as considered here, it is inferred from ergodicity that ensemble averaging is identical to volume and time averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' As a consequence, Dconv is computed from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='4) with the ensemble average being replaced in practice by a volume average combined with a time average over an appropriate time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The numerical methods employed to solve the two-phase flow have been described in detail in Loisy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' In short, we employ a standard projection method (Chorin 1968) to integrate the incompressible Navier-Stokes equations, a level-set method (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=', (Sussman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 1994)) to capture the moving gas-liquid interface, and surface tension is accounted for using the continuum surface force model (Brackbill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (2014) using a third-order total-variation- diminishing (TVD) Runge-Kutta scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The level-set function is then reinitialized using the procedure of Russo & Smereka (2000), and a correction is finally applied to enforce volume conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (ii) The scalar transport equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='3) is advanced by using a mixed Crank- Nicolson/third-order Adams-Bashforth time-stepping scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Loisy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Naso, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Spelt case Bo Ar Nb φ Re bubble shape S0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='15 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='00164 spherical S1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='38 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='03 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='002 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='72 spherical C 243 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='002 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='44 skirted E1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='0 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='9 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='002 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='9 ellipsoidal E1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='0 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='9 [1, 12] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='024 ≈ 30 ellipsoidal Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The resulting bubble Reynolds number (Re) and shape are also provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Spatial discretization relies on a mixed finite difference/finite volume approach on a fixed, staggered, Cartesian grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Second-order centered schemes are generally employed, except for advective terms which are discretized using fifth-order weighted-essentially- nonoscillatory (WENO) schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Results of numerical tests are presented in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Parametric study Four different flow regimes, as defined by the set (Ar, Bo), are considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' These are described in table 1, and have been studied in Loisy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (2017) (the same case code names are used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' In case S0, the bubbles are spherical and the Reynolds number is vanishingly small, which approaches Stokes flow conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' In case S1, the bubbles are (nearly) spherical and Re ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' In case C, the bubbles are skirted, and Re ≈ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' In case E1, the bubbles are ellipsoidal, and Re ≈ 30 − 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Ordered arrays of bubbles in these four flow regimes have been considered for the smallest volume fraction numerically accessible (value provided in table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Simulations of scalar transport in free arrays have been performed for 2 ⩽ Nb ⩽ 12 in case E1 at φ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='4 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' For each of these configurations (Ar, Bo, φ, Nb), the drift velocity (and thereby the Reynolds number) is known from Loisy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' This allowed us to impose the P´eclet number a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Ordered suspensions We examine in this section the dispersion of a passive scalar in ordered suspensions of deformable bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Our main objective here is to elucidate the effects of inertia on dispersion, using theoretical analysis and numerical simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The effective diffusivity of ordered and freely evolving bubbly suspensions 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Asymptotic analysis We first determine analytically the convective contribution to the effective diffusivity of ordered suspensions of spherical fluid particulates (bubbles or drops).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The Reynolds number of the particulates is assumed to be small so that the Navier-Stokes equations can be approximated by the Oseen equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='1) where h is the lattice spacing and ei are the unit vectors aligned with the primitive axes of the cubic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' In the dilute limit (db/h ≪ 1), the action of these particulates on the fluid can be represented by point forces −f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The convective contribution to the effective diffusivity arising from the far field has been derived by Koch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (1989) for an ordered array of rigid spheres in the Stokes flow regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' In what follows we extend their result to the case of spherical fluid particulates at small but finite Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' When Pe ≪ 1, the convective contribution to the effective diffusivity arising from the far field can be approximated by (Koch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 1989): Dconv D = � k̸=0 k2ˆu′(k)ˆu′(−k) (2π)2k4D2 + (U · k)2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='2) where the summation is over all vectors k in the reciprocal lattice k = 1 h (n1e1 + n2e2 + n3e3) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='3) and where ˆu′ is the three-dimensional Fourier transform of the velocity disturbance u′ = u − ⟨u⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='4) where f is the hydrodynamic force exerted by the ambient fluid on a particulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' In the dilute limit, f can be approximated by the Oseen drag exerted on a single spherical fluid particulate: f = Ff 0,Stokes (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='5) where f 0,Stokes is the Stokes drag on that particulate (Hadamard 1911;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Rybczynski 1911): f 0,Stokes = −2πµ∗µcdbU, with µ∗ = µc + 3µd/2 µc + µd , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='6) and where F accounts for the finite-Re correction to the Stokes drag (Brenner & Cox 1963): F = 1 + 1 8µ∗Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='8a) 10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Loisy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Naso, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='8), derived for an ordered array of point particulates in Oseen flow conditions (F is the Oseen drag divided by the Stokes drag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='8b) with U ∗ = U/U, k∗ = kh, Reh = ρcUh/µc, and Peh = Uh/D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The solution given by Koch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (1989) (equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='5) therein) for rigid spheres and Stokes flow is recovered in the limit Re → 0 and µd/µc → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' As highlighted by Koch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The asymptotic behavior of ∥Dconv∥, where ∥ ∥ denotes the tensorial Frobenius norm, is provided in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=', Loisy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (2017)), we restrict the analysis to the simplest case of bubbles rising vertically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='10) Note that F can also be expressed in terms of commonly employed dimensionless groups: F = 1 12µ∗ Ar 2 Re .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Note that in (a), Dconv ∥ is compensated by Pe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='7) when φ → 0 and when Oseen-flow approximation is applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The longitudinal and transverse components of the convective contribution, Dconv ∥ and Dconv ⊥ respectively, have been calculated from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='8) for db/h = 10−6 as a function of Peh for various Re < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The results, shown in figure 1, indicate that the asymptotic dependences of Dconv ∥ and Dconv ⊥ on Pe are independent of Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The sole effect of inertia is to modify the proportionality constants (by a substantial amount for the transverse component though).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' In this regime, diffusion is much faster than convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' This regime corresponds to the “convectively enhanced dispersion” regime in Koch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' We emphasize that the expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='8) has been derived from the approximation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='2), the validity of which is established only for Pe ≪ 1 (which is, in practice, of limited use).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Using symmetry arguments, Koch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (1989) (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Loisy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Naso, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Spelt Peh D|| conv (D F2 Pe2) x103 100 101 102 103 104 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='8 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='6 (a) Re = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='0, spherical (S0) Re = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='7, spherical (S1) Re = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='2 %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The normalizations of Dconv ∥,⊥ are those suggested by the asymptotic analysis (identical to those used in figure 1), and F is given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The lines are drawn to guide the eyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Note that in (a), Dconv ∥ is compensated by Pe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' x+he3, so that dispersion can only occur if diffusion across streamlines is present (Koch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' In the presence of inertial effects, the reflectional symmetry is lost, hence this argument does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Koch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='3 therein, note that their Pe corresponds to Peh in our notations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Such a demonstration for Oseen flow will not be attempted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Instead, the range Pe ⩾ 1 will be explored using direct numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Numerical results The above analysis provides explicit expressions of Dconv ∥ and Dconv ⊥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' We shall now determine using numerical simulations whether these restrictions can be relaxed, and if so, to which extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The longitudinal and transverse components of the convective contribution to the effective diffusivity have been computed for h/db = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='4, which corresponds to a gas volume fraction of φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='1 and figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The different colors, symbols and line styles depict the different flow regimes (the lines are drawn to guide the eyes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='95 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='2 (a) Re = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='0, spherical (S0) Re = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='7, spherical (S1) Re = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='2 %): longitudinal (a) and transverse (b) components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Dconv,anal is given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (Dconv ∥,⊥ ∝ DPe2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Inertial effects and bubble deformation only affect the proportionality constants, rather weakly for Dconv ∥ but substantially for Dconv ⊥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='8) with F computed directly from its definition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 1(a) and 2(a), this quantity does not vary more than 15% for the cases studied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' We conclude that, at small volume fraction, Dconv ∥ can be predicted within ±5 % from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='8) at any P´eclet number up to 103 and any Reynolds number up to 40, even when the bubbles are strongly deformed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' For the transverse component, the asymptotic analysis underpredicts the value of Dconv ⊥ at high P´eclet number, even for Re ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' As a consequence, Dconv ⊥ cannot be accurately estimated from our analytical solution when the assumptions underlying its derivation are not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Loisy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Naso, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Spelt Re = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='0 (S0) Pe = 10−1 Re = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='7 (S1) Re = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='4 (C) Re = 40 (E1) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='00 c’ Pe = 103 −300 −200 −100 0 c’ Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The imposed scalar field ¯c increases linearly within the cell from bottom to top (φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='2 %, the entire cell is shown, and gravity is pointing downward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The effective diffusivity of ordered and freely evolving bubbly suspensions 15 Re = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='0 (S0) Pe = 10−1 Re = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='7 (S1) Re = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='4 (C) Re = 40 (E1) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='002 c’ Pe = 103 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='4 c’ Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The imposed scalar field ¯c increases linearly within the cell from left to right (φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='2 %, the entire cell is shown, and gravity is pointing downward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 16 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Loisy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Naso, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Symbols other than purple stars: DNS (Re ≈ 30, φ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='4 %);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' purple stars: experimental data of Alm´eras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (2015) (Re ≈ 700, φ ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='4 %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Dconv ∥ and Dconv ⊥ , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The field of c′ associated with Dconv ∥ , shown in figure 4, exhibits similar features at low and high Pe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' In contrast, the field of c′ associated with Dconv ⊥ , represented in figure 5, is qualitatively different in these two limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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 ⊥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='4 %) and explored the effect of varying the number of free bubbles Nb on the dependence of Dconv on the P´eclet number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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 ⊥ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' For the same reason, investigations of the effects of volume fraction and flow regime could not be undertaken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Instantaneous scalar fluctuation field c′ associated with Dconv ∥ for a free array of 8 bubbles, at Pe = 10−1 (left) and Pe = 106 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The gradient of ¯c is vertical (the entire cell is shown, and gravity is pointing downward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Instantaneous scalar fluctuation field c′ associated with Dconv ⊥ for a free array of 8 bubbles, at Pe = 10−1 (left) and Pe = 106 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The gradient of ¯c is horizontal (the entire cell is shown, and gravity is pointing downward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' function of the P´eclet number for various values of Nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Note that a very wide range of P´eclet numbers is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' We first examine the dependence of Dconv on the P´eclet number in free arrays of bubbles (Nb > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' At small Pe, Dconv ∥,⊥ ∝ DPe2, whereas at high Pe, Dconv ∥,⊥ ∝ DPe = Udb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Note that the scaling at high Pe is expected from a simple dimensional analysis in a convection-dominated regime where diffusion plays no role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' This regime corresponds to the “mechanical dispersion” regime in Koch & Brady (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The different dispersion regimes at low and high Pe can also be identified from the features of the scalar fluctuation field c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Instantaneous snapshots of c′ associated with Dconv ∥ and Dconv ⊥ are 18 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Loisy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Naso, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' For a given component, the isocontours of c′ follow markedly different patterns at low and high Pe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' We now compare these results with those obtained for ordered arrays (black crosses in figure 6) and discuss the effect of the microstructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' At small Pe, Dconv ∥ and Dconv ⊥ grow quadratically with Pe in both free and ordered arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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 ⊥ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Loisy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' In this regime, the transverse dispersion is indeed governed by mechanical dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Incidentally, mechanical dispersion is also obtained at high Pe in random media in Stokes flow conditions (Koch & Brady 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Although the microstructure of the present bubbly suspensions has not been evaluated quantitatively, visual inspection and prior results on their dynamics (Loisy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 2017) showed that it is not random, but rather characterized by a certain “organization”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' We stress that this last statement is purely speculative, and would require a quantitative study of the effect of the microstructure to be confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' We finally attempt a comparison of our results with the experimental data of Alm´eras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (2015), who measured the effective diffusivity of a homogeneous swarm of high- Reynolds-number rising bubbles at Pe ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='75×106 for gas volume fractions ranging from 1 % to 13 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' It is important to stress that in these experiments, Re ≈ 700, whereas in the simulations, Re ≈ 30, so the comparison is only indicative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Interpolation (by eye) of their data at φ ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='4 % (figure 10 in their paper) yields Deff ∥ /D = 1×106 and Deff ⊥ /D = 5×105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' These experimental values are represented by purple stars in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (2015)), diffusivity ratio, and bubble-induced liquid agitation in the horizontal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Finally, as only one value of the P´eclet number was considered in the experiments of Alm´eras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (2015), no comparison of their data with our results can be offered regarding the dependence of the effective diffusivity on the P´eclet number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Conclusions In this study we investigated scalar dispersion in homogeneous bubbly suspensions as described by an effective diffusivity tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' This convective contribution is that associated with bubble-induced agitation, and is the dominant contribution to the effective diffusivity in commonly encountered bubbly flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The dispersion theory of Koch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (1989) indicates that convective mixing mech- anisms in ordered suspensions in Stokes-flow conditions differ at low and high P´eclet numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' This result has been confirmed by direct numerical simulations for values of the Reynolds number ranging from vanishingly small to moderate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' At low P´eclet number, dispersion in free arrays is convectively enhanced, as in ordered ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' This suggests that the limiting role of molecular diffusion to Taylor dispersion is taken over by mechanical dispersion, or that mechanical dispersion itself dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Besides, the effective diffusivity seems to be weakly sensitive to the number of bubbles present in a unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Loisy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Naso, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Spelt interface, and therefore cannot be straightforwardly compared to those obtained in real bubbly flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='5) in addition to the convective one considered in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Jeffrey 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Koch & Brady 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Heat and mass exchanges across phase boundaries are traditionally expressed as dimensionless transfer coefficients called the Nusselt and the Sherwood numbers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Their functional dependences on suspension properties, in particular the volume fraction, have been the subject of analytical (Acrivos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 1980), numerical (Aboulhasanzadeh & Tryggvason 2014), and experimental (Colombet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 2011, 2015) studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' They correspond to different closure problems, and one cannot be inferred from the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Nevertheless, the present work will be primarily important for mass transfer processes in bubbly flows that are liquid-phase controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Appendix: Spatial convergence tests We present the results of some spatial convergence tests of the algorithm solving the scalar transport equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The results of similar tests for the algorithm solving the flow are shown in Loisy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Loisy (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='4 %), in both ordered and free configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' For ordered arrays, three different resolutions were tested, namely db/∆x = {20, 40, 60} with ∆x the grid spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' The results are shown in figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' This resolution is the same as that used for the simulation of the corresponding bubbly flow in Loisy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' (2017)), The effective diffusivity of ordered and freely evolving bubbly suspensions 21 log10 (∆x db) −2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='8 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='6 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='4 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' namely 60 grid cells per diameter for case C and 40 grid cells per diameter for the other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' Simulations at higher resolution were too expensive to be continued over sufficiently long times to allow a quantitative estimate of the uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' A resolution of 20 grid cells per diameter was therefore concluded to be sufficient for free arrays in view of the present purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' For a given case, the same resolution was used for all P´eclet numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' REFERENCES Aboulhasanzadeh, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' & Tryggvason, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' 2014 Effect of bubble interactions on mass transfer in bubbly flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AyT4oBgHgl3EQfPvYS/content/2301.00028v1.pdf'} +page_content=' International Journal of Heat and Mass Transfer 79, 390–396.' metadata={'source': 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+Comprehensive Survey and Future directions +*Corresponding author(s) +1st Teerath Kumar* +ADAPT - Science Foundation Ireland +Research Centre and CRT AI, +School of Computing, Dublin City University, +Dublin, Ireland; +teerath.menghwar2@mail.dcu.ie +2nd Muhammad Turab +Department of Computer Systems +Engineering Mehran University of Engineering +and Technology Jamshoro, Pakistan; +turabbajeer202@gmail.com +3rd Kislay Raj +CRT AI, School of Computing, +Dublin City University, +Dublin, Ireland; +kislay.raj2@mail.dcu.ie +4th Alessandra Mileo +INSIGHT Centre for Data Analytics +and the I-Form Centre for Advanced Manufacturing, +School of Computing, +Dublin City University, Ireland; +alessandra.mileo@dcu.ie +5th Rob Brennan +ADAPT, School of Computer Science, +University College Dublin, +Ireland; +rob.brennan@adaptcentre.ie +6thMalika Bendechache +ADAPT & Lero Research Centres, +School of Computer Science, +University of Galway, Galway, Ireland; +malika.bendechache@universityofgalway.ie +Abstract—Deep learning (DL) algorithms have shown signif- +icant performance in various computer vision tasks. However, +having limited labelled data lead to a network overfitting prob- +lem, where network performance is bad on unseen data as +compared to training data. Consequently, it limits performance +improvement. To cope with this problem, various techniques +have been proposed such as dropout, normalization and ad- +vanced data augmentation. Among these, data augmentation, +which aims to enlarge the dataset size by including sample +diversity, has been a hot topic in recent times. In this article, +we focus on advanced data augmentation techniques. we provide +a background of data augmentation, a novel and comprehensive +taxonomy of reviewed data augmentation techniques, and the +strengths and weaknesses (wherever possible) of each technique. +We also provide comprehensive results of the data augmentation +effect on three popular computer vision tasks, such as image +classification, object detection and semantic segmentation. For +results reproducibility, we compiled available codes of all data +augmentation techniques. Finally, we discuss the challenges and +difficulties, and possible future direction for the research commu- +nity. We believe, this survey provides several benefits i) readers +will understand the data augmentation working mechanism to fix +This publication has emanated from research [conducted with the financial +support of +supported in part by a grant from] Science Foundation Ireland +under Grant number 18/CRT/6223 and is supported by the ADAPT Centre for +Digital Content Technology which is funded under the SFI Research Centres +Programme (Grant 13/RC/2106/P2), Lero SFI Centre for Software (Grant +13/RC/2094/P2) and is co-funded under the European Regional Development +Fund. For the purpose of Open Access, the author has applied a CC BY +public copyright licence to any Author Accepted Manuscript version arising +from this submission +overfitting problems ii) results will save the searching time of the +researcher for comparison purposes. iii) Codes of the mentioned +data augmentation techniques are available at 1 iv) Future work +will spark interest in research community. +Index Terms—Big data, Computer vision, Data Augmenta- +tion,Deep learning, Image classification, Object detection, Seman- +tic segmentation, Survey Data Augmentation +I. INTRODUCTION & MOTIVATION +Deep learning models have been very popular and made +immense progress in computer vision (CV) tasks such as im- +age classification [11], [48], [60], [68], [70], [71], [103], [110], +object detection [40], [47], and image segmentation [74], [79], +[81], [86]. All this advancement has been accelerated by +different deep neural network architectures, powerful compu- +tation resources, a large amount of accessible data, and mature +deep learning libraries. Among the deep learning models, +Convolution Neural Networks (CNNs) have performed well on +computer vision tasks. CNNs apply the convolution operation +with the input image and kernel to learn different features +in an image. The initial layers of CNN learn the low-level +features (i.e edges, lines, etc) while the deep layers learn +more structured complex features. The success of CNN has +caught the attention to apply it for computer vision tasks. +Along with CNN, the Vision Transformers (ViT) [28] are also +1https://github.com/kmr2017/Advanced-Data-augmentation-codes +arXiv:2301.02830v1 [cs.CV] 7 Jan 2023 + +getting popular and have been widely used in deep learning for +computer vision tasks. Although these algorithms are popular +and have shown excellent performance in deep learning, they +require a lot of data to learn the correct features and avoid +overfitting problem [104]. Overfitting is when a model is +performing well on training data but is not performing on +the test (unseen) data, as shown and explained in figure 1. +However, data is not always available in large quantities due to +various reasons such as privacy issues (e.g., medical imaging +analysis) or the need for tedious human labeling (object +detection, image segmentation) etc [108]. Another reason is, +it is always tedious, time-consuming and expensive to label +data in the case of an availability of unlabeled data [71]. +Even in the availability of huge datasets such as imageNet, +data augmentation can still help to reduce overfitting effect. It +happens because, with a standard training process, the model +learns only the important regions (for example head of a +dog). But it is also necessary for the model to learn other +less important features to be more generalized +[146]. The +CNNs trained on the small set of data often lead to overfitting. +Another concern is the adversarial attacks [50], [88], [152], +where the noisy perturbation is added to the input image to +fool the CNNs and consequently degrade DNNs accuracy. This +modification caused by perturbation is invisible to the human +eyes but makes the network fail to identify the correct features +in an image. +To address these problems, data augmentation is mostly +applied. It is not only useful in computer vision tasks, but +also helpful in number of domains such as audio +[3], [12], +[65], [72], [92], [95], [111], [124] and text domains [6], [33], +[84], [109] as well. In this survey, we focus only the computer +vision domain. +Regularization is a technique that generalizes well the model +from architectural and data perspectives. There are several +forms of regularization such as Dropout [117], Batch normal- +ization [56], transfer learning [107], [137], pre-training [30], +data augmentation [154], human-in-the-loop for data aug- +mentation [10] and many others [108]. Data augmentation +is the form of regularization explicitly +[69], [110], [154]. +Technically, it enlarges the dataset by changing the sample +view or flavour [154] to give a diverse view. Other mentioned +techniques do not work directly on data like data augmentation +as the data is the main cause of any problem for training. If it +has an overfitting issue or is biased, it will be propagated to +the model as well. Carefully performing data augmentation is +the key challenge as it is discussed in section IV. Data aug- +mentation is performed on assumption that more information +can be extracted from the real dataset. But this assumption is +not true in real world scienario. +Generally, data augmentation solves two key problems. i) the +problem of lack of data or limited data, consequently it leads to +problem of overfitting. To solve the overfitting, data augmen- +tation makes the model more generalized based on scenario(s). +This can be achieved by feeding the various possible scenarios +of an image. This indicates more information is extracted +indirectly from the original dataset. ii) Labeling, the original +dataset has a label for each sample. Augmenting each sample, +the label is assigned to the augmented sample as that of the +original sample. In some augmentations, the label information +is not preserved such as in Mixup data augmentation, labels +are also mixed to augment a label. +There are numerous surveys on data augmentation. Wang +et al. +[98] explores and compares several traditional data +augmentations for image classification tasks only. In another +work, Wang et al. +[130] review available data augmenta- +tion approaches for facial data. This work is only limited +to face recognition. Khosla et al. [61] discuss warping and +oversampling-based data augmentation approaches. No taxon- +omy, no literature review and no evaluation of the methods +are discussed. Shorten et al. [108] provide a very detailed +work with different aspects of data augmentation, but they +did not provide an evaluation of the data augmentation for +different CV tasks and they did not include state-of-the-art +(SOTA) augmentation methods such as cutmix and grid mask +etc. Previously the discussed surveys have been two years old +and in the last the years, there have been proposed several data +augmentation techniques, so it is a dire need for a survey. +Recently Yang et al. [142] provides a detailed survey with +results of several computer vision tasks. Very limited results +are compiled and SOTA data augmentation methods are not +covered. Another recently work +[141] by Xu, discusses the +data augmentation that is model-based and model-free and, +proposes novel taxonomy. But this work fails to provide +the evaluation of the data augmentation and discusses very +limited data augmentations. There are the number of the +data augmentation based on generative adversarial network +(GAN) [118], [145], but we do not cover GAN-based data +augmentations in this work, as GAN is itslef vast topic and +GAN-based techniques are very huge in number Interestingly, +none of the mentioned works provides extensive evaluations +of SOTA data augmentation and available code compilation +based on the proposed taxonomy for result reproducibility. +To fill these mentioned gaps, our survey makes the following +contributions: +• Presents novel Data augmentation taxonomy. +• Explains SOTA augmentation approaches with visualiza- +tion. +• Presents SOTA augmentation evaluation for several tasks. +• Compiles the available codes of data augmentations fol- +lowing the proposed taxonomy for results reproducibility. +• This survey discusses data augmentation challenges and +future directions +• This survey provides open research questions +The above contributions provide the following benefits: +• A better understanding of data augmentation working +mechanism to fix the overfitting problem. +• Our comprehensive analysis and comparison between +the existing data augmentation techniques will save re- +searchers’ time searching this field. +• facilitate result reproducibility by providing the source +code for the different data augmentation techniques in- + +Fig. 1. Overfitting problem: On the left side, overfitting is explained in terms of accuracy, after the inflation point (red dotted line), the training accuracy is increasing but validation accuracy is decreasing. +On the right side, alternatively in terms of loss, training loss is decreasing but validation loss is increasing after the red dotted line. The figure is taken from the source 3https://www.baeldung.com/cs/ml- +underfitting-overfitting + +Accuracy +Loss +Training +Validation +Validation +Training +Epochs +Epochsvestigated. +• Future work will spark interest in the research commu- +nity. +II. TAXONOMY AND BACKGROUND +In this section, we discuss the proposed taxonomy as shown +in the figure 2, first data augmentation is classified into +two branches, i) basic data augmentations ii) Advanced data +augmentations. Then these two are classified further based on +operations. Background and explanation of each augmentation +are discussed below taxonomically: +A. Basic Data Augmentation Methods +This section describes basic data augmentation methods and +classifies the augmentation techniques. +1) Image Manipulation: Image manipulation refers to the +changes made in an image with respect to its position or color. +The positional manipulation is made by adjusting the position +of the pixels while color manipulations are made by altering +the pixel values of the image. Image manipulation is further +divided into two main categories. Each of them is discussed +below. +Geometric Data Augmentation: Geometric augmentation +refers to the changes made with respect to the image geometry. +Geometry refers to position, shifting at certain angle etc. This +technique alters the position of pixel values in image. e.g. +Rotation, Translation, and Shearing. Basic geometric augmen- +tations are shown in figure 3. +(i) Rotation : Rotation data augmentation where image is +rotated between 0 and 360 degree. Degree of rotation is a +hyperparameter, it should be chosen wisely. Like in case +MNIST we can not rotate 180 rotations, i.e. rotation 6 +digit by 180 degree, it will be 9. So it won’t make sense. +It depends on the dataset. +(ii) Translation : It is another geometric type data augmen- +tation, which shifts the image in upward, downward, right +or left direction to give diverse view. The demonstration +is shown in the second of the figure 3. +(iii) Shearing : +Word ‘shear’ means to pervert an image along an axis. +Shearing is a data augmentation technique that shifts one +part of the image to one direction, while the other part is +in the reverse direction. Technically, it is divided into two +categories, x shear and y shear. In x shear, the top part +of the image is shifted in one direction and the bottom +is shifted in the totally opposite direction. In y shear, the +left part of the image is shifted in one direction and the +right part is shifted in the reverse direction. +2) Non-Geometric Data Augmentations: This category fo- +cuses on the visual appearance of the image rather than its ge- +ometrical shape. Noise injection, flipping, cropping, resizing, +and color space manipulation is examples of non-geometric +augmentation techniques. Some examples of non-geometric +data augmentations are shown in figure 4. A few classical +approaches are discussed below. +(i) Flipping : it is a kind of data augmentation technique +that flips the image either horizontally or vertically, it +has shown positive results on the most popular datasets +such as cifar10, cifar100 [67] and many more. +(ii) Cropping and resizing : Cropping is another data aug- +mentation technique that is used as a preprocessing aug- +mentation. Either random cropping or central cropping is +used as data augmentation. This technique decreases the +size of the image then resizing is performed to match the +original size of the image, while the labels of the image +are not smoothed. +(iii) Noise Injection : Injection noise is another technique of +data augmentation, that helps neural networks to learn +robust features and is quite helpful in defending against +adversarial attacks. Nine datasets from the UCI repository +have shown impressive results (Reference from [108]). +(iv) Color Space: Images having dimensions of H x W x +C (where H, w and C represent the height, width and +channels, respectively) consist of three channels R, G +and B. Manipulating each channel values separately in +order to control brightness is another way of data aug- +mentation, sometimes it is also referred as photometric +augmentation. This augmentation is useful for avoiding +the model to be biased toward lightning conditions. The +Simplest way of performing color space augmentation +is to isolate any channel and add 2 channels filled with +any random value or 0 or 255. Color space is used in +photo editing applications i.e. to control the brightness +or darkness [108]. +(v) Jitter: It is another data augmentation technique, that +randomly changes the brightness, contrast and saturation +and hue of the image. These four are the hyperparameters +and their range (min-max) should be chosen carefully. For +example, if we increase the brightness of X-Ray images +for lung disease detection, it will whiten and mix the lung +in X-ray and won’t help disease diagnosis. (examples will +be shown). +(vi) Kernel Filters: it is another data augmentation technique +that sharpens or blurs the image. It starts first, we slide +the window of size n x n kernel/matrix of gaussian blur +filter or edge filter. Gaussian blur filter blurs the image +and the edge filter sharpens the edge of the image either +horizontally or vertically. +3) Image Erasing Data Augmentations: +a) Cutout: It randomly erases the sub region and fills +with 0 or 255 in an image during training. It shows the +impressive performance on very popular datasets [27]. The +demonstration of cutout is shown in figure 16. +b) Random erasing: It [154] randomly erases the sub +region in the image like a cutout. But it also randomly deter- +mines to mask out or not and also determines the aspect ratio +and size of the masked region. Random erasing demonstration +for different tasks is shown in figure 5. +c) Hide-and-Seek: The key idea of hide-and-seek data +augmentation [114] is to divide the image into uniformly +squares of random size and randomly remove a random + +Data Augmentations +Basic Data Augmentations +Advanced Data Augmentations +Image Manipulation +Image Erasing +Image Mixing +Auto Augment +Neural Style Transfer +Feature Augmentation +Single Image Mixing +• Local Augment +• SalfMix +• AutoAugment +• Fast AutoAugment +Multi-Image Mixing +• Mixup +• CutMix +Geometric +Manipulation +• Flipping +• Cropping +Non-Geometric +Manipulation +• Rotation +• Translation +Reinforcement +Learning Based +Non-Reinforcement +Learning Based +• FeatMatch +• Adversarial +Feature Aug +Erasing +• Cutout +• GridMask +• RandAug +Neural Style +• Style Aug +• StyPath +Feature Aug +Fig. 2. Proposed image data augmentation taxonomy + +Fig. 3. Overview of the geometric data augmentations. +number of squares. It forces neural networks to learn relevant +features when important information is hidden. At each epoch, +it gives a different view of an image as shown in figure 6. +d) GridMask Data Augmentation: GridMask [15] ad- +dresses the problem of randomly removing regions that either +can completely erase the object or remove context informa- +tion. To do a trade-off between these problems, GridMask is +proposed by creating uniform masking and then applying it to +images. It is shown in the figure 7. +B. Image Mixing Data Augmentations +Image mixing data augmentation has been a hot topic for +the last few years. Image mixing data augmentation is about +mixing image(s) with others or the the same image(s). In this +work, we classify the image mixing data augmentation into +two categories: +• Single image mixing +• Non-single image mixing +• Single Image Mixing Data Augmentations +The single image mixing technique uses only one image +and plays around with it from different strategic points of +view. Recently there has been a lot of work done on single- +image augmentation, such as LocalAugment, SelfAugmenta- +tion, SalfMix, etc. The description of each SOTA single image +mixing data augmentation has been discussed below. +(i) Local Augment: This paper [64] proposes a local aug- +ment, that divides image into patches and applies dif- +ferent kinds of data augmentation on each with the aim +of potentially changing bias properties but generating +significant local features, as shown in the figure below. +Fig. 4. Overview of the non-geometric data augmentations. +Fig. 5. Random erasing examples for different tasks. Figure source is [154] + +Input +Augmentation +Output +Rotation +Translation +ShearingInput +Augmentation +Output +Flipping +Cropping & Resize +Noise Injection +Color Spacing +Color Jitterimage classification +person re-ID +input image +Random ErasingAccuracies +Method +CIFAR10 +CIFAR10+ +CIFAR100 +CIFAR100+ +ResNet-18 (Baseline) +89.37 +95.28 +63.32 +77.54 +ResNet-18 + CutOut +90.69 +96.25 +65.02 +80.58 +ResNet-18 + Random Erasing +95.28 +95.32 +- +- +ResNet-18 + CutMix +90.56 +96.22 +65.58 +80.58 +ResNet-18 + SaliencyMix +92.41 +96.35 +71.27 +80.71 +ResNet-18 + GridMask +95.28 +96.54 +- +- +ResNet-50 (Baseline) +87.86 +95.02 +63.52 +78.42 +ResNet-50 + CutOut +91.16 +96.14 +67.03 +78.62 +ResNet-50 + CutMix +90.84 +96.39 +68.35 +81.28 +ResNet-50 + SaliencyMix +93.19 +96.54 +75.11 +81.43 +WideResNet-28-10 (Baseline) [125] +93.03 +96.13 +73.94 +81.20 +WideResNet-28-10 + CutOut [27] +94.46 +96.92 +76.06 +81.59 +WideResNet-28-10 + Random Erasing +96.2 +96.92 +81.59 +82.27 +WideResNet-28-10 + GridMask +96.13 +97.24 +- +- +WideResNet-28-10 + CutMix +94.82 +97.13 +76.79 +83.34 +WideResNet-28-10 + PuzzleMix +- +- +- +83.77 +WideResNet-28-10 + SaliencyMix +95.96 +97.24 +80.55 +83.44 +Note: + sign after dataset name show that traditional data augmentation methods have been used +TABLE I +BASELINE PERFORMANCE COMPARISON OF VARIOUS AUGMENTATION ON CIFAR10 AND CIFAR100 DATASETS. +Fig. 6. +An example of Hide-and-Seek augmentation, image is taken from +[114] +Though this augmentation does not main global structure +but provides very diverse features of images, that are +essential for neural networks to learn local features in a +more generalised way. The visual representation is shown +in figure 8 and 9. +(ii) Self Augmentation: This paper [106] proposes the self- +augmentation, where a random region of an image is +cropped and pasted randomly in the image, improves +the generalization capability in few-shot learning. The +process demonstrated in the figure 10. +(iii) SalfMix: This paper [20] focuses on whether it is pos- +sible to generalize neural networks based on single- +image mixed augmentation? For that purpose, it proposes +SalfMix, the first salient part of the image is found +to decide which part should be removed and which +Fig. 7. +This figure shows the procedure of GridMask augmentation. They +produce a mask and then multiply it with the input image, the image is taken +from [15]. +portion should be duplicated. Most salient regions are +cropped and placed into non-salient regions. This process +is defined and compared with other techniques in the +figure 11. +(iv) KeepAugment KeepAugment [41] is introduced to pre- +vent distribution shift which degrades the performance +of neural networks. KeepAugment’s idea is to increase + +Training phase +W +Epoch 1 +CNN +H +Epoch 2 +CNN +s +Training image +EpochN +CNNCIFAR-10 +CIFAR-100 +ImageNet +Augmentation +Accuracy (%) +Model +Accuracy (%) +Model +Accuracy (%) +Model +Cutout [27] +97.04 +WRN-28-10 +81.59 +WRN-28-10 +77.1 +ResNet-50 +Random Erasing [154] +96.92 +WRN-28-10 +82.27 +WRN-28-10 +- +- +Hide-and-Seek [114] +95.53 +ResNet-110 +78.13 +ResNet-110 +77.20 +ResNet-50 +GridMask [15] +97.24 +WRN-28-10 +- +- +77.9 +ResNet-50 +LocalAugment [64] +- +- +95.92 +WRN-22-10 +76.87 +ResNet-50 +SalfMix [20] +96.62 +PreActResNet-101 +80.11 +PreActResNet-101 +- +- +KeepAugment [41] +97.8 +ResNet-28-10 +- +- +80.3 +ResNet-101 +Cut-Thumbnail [140] +97.8 +ResNet-56 +95.94 +WRN-28-10 +79.21 +ResNet-50 +MixUp [147] +97.3 +WRN-28-10 +82.5 +WRN-28-10 +77.9 +ResNet-50 +CutMix [146] +97.10 +WRN-28-10 +83.40 +WRN-28-10 +78.6 +ResNet-50 +SaliencyMix [125] +97.24 +WRN-28-10 +83.44 +WRN-28-10 +78.74 +ResNet-50 +PuzzleMix [63] +- +- +84.05 +WRN-28-10 +77.51 +ResNet-50 +FMix [45] +98.64 +Pyramid +83.95 +Dense +77.70 +ResNet-101 +MixMo [101] +96.38 +WRN-28-10 +82.40 +WRN-28-10 +- +- +StyleMix [52] +96.44 +PyramidNet-200 +85.83 +PyramidNet-200 +77.29 +PyramidNet-200 +RandomMix [85] +98.02 +WRN-28-10 +84.84 +WRN-28-10 +77.88 +WRN-28-10 +MixMatch [9] +95.05 +WRN-28-10 +74.12 +WRN-28-10 +- +- +ReMixMatch [8] +94.71 +WRN-28-2 +- +- +- +- +FixMatch [115] +95.69 +WRN-28-2 +77.04 +WRN-28-2 +- +- +AugMix [49] +- +- +- +- +77.6 +ResNet-50 +Improved Mixed-Example [120] +96.02 +ResNet-18 +80.3 +ResNet-18 +- +- +RICAP [122] +97.18 +WRN-28-10 +82.56 +ResNet-28-10 +78.62 +WRN-50-2 +ResizeMix [100] +97.60 +WRN-28-10 +84.31 +WRN-28-10 +79.00 +ResNet-50 +AutoAugment [23] +97.40 +WRN-28-10 +82.90 +WRN-28-10 +83.50 +AmoebaNet-C +Fast AutoAugment [82] +98.00 +SS(26 2×96d) +85.10 +SS(26 2×96d) +80.60 +ResNet-200 +Faster AutoAugment [46] +98.00 +SS(26 2 × 112d) +84.40 +SS(26 2×96d) +75.90 +ResNet-50 +Local Patch AutoAugment [83] +98.10 +SS(26 2 × 112d) +85.90 +SS(26 2×96d) +81.00 +ResNet-200 +RandAugment [24] +98.50 +PyramidNet +83.30 +WRN-28-10 +85.00 +EfficientNet-B7 +TABLE II +PERFORMANCE COMPARISON OF THE VARIOUS IMAGE ERASING AND IMAGE MIXING AUGMENTATIONS FOR IMAGE CLASSIFICATION PROBLEMS. WRN +STANDS FOR WIDERESNET AND SS FOR SHAKE-SHAKE. +Fig. 8. An example of Global and Local Rotation Image, example is taken +from [64]. +Fig. 9. Comparison of LocalAugment with CutOut, MixUp etc, example is +taken from [64]. +fidelity by preserving the salient features of the image +and augmenting the non-salient region. Preserved features +further allow for increased diversity without shifting the +distribution. Keep augment is shown in the figure 12. +(v) You Only Cut Once You Only Cut Once (YOCO) +[44] is introduced with the aim of recognizing objects +from partial information and improving the diversity of +augmentation that encourage neural networks to perform + +Input Image +GlobalRotation +1 +LocallySegmented +LocalRotation +Input ImageInput Image +Cutout +Random Erasing +Mixup +CutMix +Local AugmentFig. 10. An example of self augmentation, image is taken from [106] +Fig. 11. Conceptual comparison between SalfMix method and other single +image-based data augmentation methods, example is taken from [20]. +Fig. 12. +This image shows the example of KeepAugment with other +augmentations, courtesy [41]. +better. YOCO makes two pieces of image and augmen- +tation is applied one each piece, then each piece is +concatenated for an image and YOCO shows impressive +performance and compared with SOTA augmentations, +sometimes it outperforms them. It is easy to implement, +has no parameters, and is easy to use. The YOCO +augmentation process is shown in the figure 13. +Fig. 13. An example of YOCO augmentation, image is taken from [44]. +(vi) Cut-Thumbnail : Cut-Thumbnail [140] is a novel data +augmentation, that resizes the image to a certain small +size and then randomly replaces the random region of +the image with the resized image, aiming to alleviate +the shape bias of the network. The advantage of Cut- +thumbnail is, that it not only preserves the original image +but also keeps it global in the small resized image. On Im- +ageNet, it shows impressive performance using resnet50. +Overall, the cut-thumbnail process and its comparison are +shown in figure 15 and figure 14, respectively. +Fig. 14. Comparison between existing data augmentation methods with Cut- +Thumbnail, example is from [140]. +Fig. 15. +This image shows an example of reduced images that is called +thumbnails. After reducing the image to a certain size 112×112 or 56×56, The +dog is still recognizable even though lots of local details are lost, courtesy +[140]. +• +Non-Single Image Mixing Data Augmentations +Non-Single image mixing data augmentation uses more than +one image and applies different mixing strategies. Recently, +many researchers explored a lot of non-single image mixing +strategies and still, it is a very attentive topic for many +researchers. Recently work has included Mixup, CutMix, + +Scale +Crop +Flip +Cutout + +Single +Training Image +Saliency Map +Self-mixed Image +(al) Red fox +(a2) Cutout +(a3) RandAugment +(a4) Saliency map +(a5) Keep+Cutout +(a6) Keep+RandAugment>S +Aug +Concat +Cut +Aug(a) Original Sample +(b) Cutout +(c) Mixup +(d) CutMix +(e) Cut-Thumbnail224SaliencyMix, and many more. Each of the relevant non- +single image mixing data augmentation techniques is discussed +below. +(i) Mixup: It blends any random two images based on the +blending factor (alpha) and the corresponding labels of +these images are also mixed in the same way. Mixup +data augmentation [147] sustainable improved the perfor- +mance not only in terms of accuracy but also in terms of +robustness. Experiments on ImageNet-2012, CIFAR-10, +CIFAR-100, Google commands and UCI datasets showed +impressive results on SOTA methods. As it is compared +and shown in the figure 16. +(ii) CutMix : It [146] tackles the issues of information loss +and region dropout issue. It is inspired by cutout [27], +where any random region is filled with 0 or 255, while in +cutmix instead of filling the random region with 0 or 255, +the region is filled with a patch from another image. Cor- +respondingly their labels are also mixed proportionally to +the number of pixels mixed. (as shown in figure 16) +Fig. 16. Overview of the Mixup, Cutout, and CutMix, example is from [146]. +(iii) SaliencyMix : It [125] basically addresses the problem +of cutmix and argues that filling a random region of the +image with a patch from another won’t guarantee that +patch has rich information and thereby mixing labels of +unguaranteed patches leads the model to learn unneces- +sary information about the patch. To deal with that issue, +saliencyMix first selects the salient part of the image and +pastes it to a random region or salient or non-salient of +another image. (as shown in figure 17 and figure 18) +Fig. 17. +An example of SaliencyMix augmentation, image is taken from +[125]. +(iv) Puzzle Mix : This article [63] proposes a puzzle mix data +augmentation technique that focuses on using explicitly +salient information and basic statistics of image wisely +with the aim of breaking misled supervision of neural +networks over existing data augmentations. Furthermore, +the demonstration is shown and compared with relevant +methods in the figure 19. +(v) SnapMix: +The article [53] proposes the Semantically +Proportional Mixing (SnapMix) that utilises class activa- +Fig. 18. +This image shows the proposed SaliencyMix data augmentation +procedure, courtesy [125] +Fig. 19. A visual comparison of the mixup methods. Puzzle Mix ensures to +contain sufficient target class information while preserving the local statistics +of each in, example is from [63]. +tion map (CAM) to reduce the label noise level. SnapMix +creates the target label considering the actual salient +pixel taking part in the augmented image, which ensures +semantic correspondence between the augmented image +and mixed labels. The overall process is demonstrated +and compared with closely matching augmentations in +the figure 20. +Fig. 20. A visual Comparison of Mixup, CutMix, and SnapMix. The figure +gives an example where SnapMix’s generated label is visually more consistent +with the mixed image’s semantic structure comparing to CutMix and Mixup, +courtesy [53]. +(vi) FMix: +This article proposes the FMix [45], a kind of +mixed sample data augmentation (MSDA), utilises the +random binary masks. These random binary masks are +acquired by applying a threshold to low-frequency images + +Target Image +Source Image +Augmented Image +20% +Mixed label for randomly mixed images +Dog - 80% & Cat 20% ? +D0g - 80% & Cat 20% ?Selecting the Peak +Selecting the +Mixing the Source +Source Image +Saliency Map of +Salient Region of +Source Patch Based +Target Image +Patch with the +Augmented Image +the Source Image +on the Peak Salient +the Saliency Map +Target Image +RegionInput1 +Input Mixup +Puzzle Mix (z only) +Input2 +CutMix +Puzzle Mix (full)MixUp +SnapMix +0.4x +0.6x +Ya: 0.4, Yb: 0.6 +Asymmetrical Mixture +Ya +Yb +Ya +CutMix +Ya: 0.6, Yb: 0.4 +Ya +Ya: 0.28, Yb: 0.42 +0.42 +0.72 +Semantic-relatedness Proportionthat are obtained from Fourier space. Once the mask is +obtained, one colour region is applied on input one and +another colour region is applied on another input. The +overall process is shown in figure 21: +Fig. 21. +Example masks and mixed images from CIFAR-10 for FMix, +example is from [45]. +(vii) MixMo : +This paper [101] focuses on the learning of +multi-input multi-output via subnetwork. Main motivation +of the paper is to replace direct hidden summing oper- +ations with more solid mechanisms. For that purpose, it +proposes MixMo, which embeds M inputs into shared +space, mixes them and passes them to a further layer for +classification. Moreover, the overall process is demon- +strated in figure 22: +Fig. 22. This image shows the overview of MixMo augmentation, image is +taken from [101]. +(viii) StyleMix : This paper [52] targets previous approaches +problems, they don’t differentiate between content and +style features. To remedy this, this problem proposes two +approaches styleMix and StyleCutMix, this is the first +work that separately deals with content and style features +of images very carefully and it showed impressive per- +formance on a popular benchmark datasets. The overall +process is defined and compared with SOTA approaches +in the figure 23: +(ix) RANDOMMIX : This paper [85] improves generaliza- +tion capability by proposing randomMix, which randomly +selects mix augmentation from a set of augmentations and +applies it to images, enabling the model to look at diverse +Fig. 23. A Visual comparison of StyleMix [52] and StyleCutMix with Mixup +[147] and CutMix [146], example is from [52]. +samples. This method showed impressive results over +SOTA image mixing methods. The overall demonstration +is shown in the figure 24: +Fig. 24. An illustrative example of RandomMix, image is taken from [85]. +(x) MixMatch : Data augmentation technique is very useful +in semi-supervised learning. MixMatch [9] augments +single image K time and passes all K number of images +to a classifier, averages their prediction and finally, their +predictions are sharpened by adjusting their distribution +temperature term. (as shown in the figure 25) +(xi) REMIXMATCH : +This work [8] is an extension of +mix match and makes prior work efficient by introduc- +ing distribution alignment and augmentation anchoring. + +Mask +Image 1 +Image 2 +FMixNetwork +Λy +Co +do +MixMo +Mixing +d +Sum Mixing (MIMO: 83.06 %) +Patch Mixing (Our Cut-MixMo: 85.40 %) +orInput 1 +Input 2 +Mixup +StyleMix +CutMix +StyleCutMix +Method +Content +Parrot 0.5 +Parrot 0.4 +Parrot 0.2 +Parrot 0.2 +label +Panda 0.5 +Panda 0.6 +Panda 0.8 +Panda 0.8 +Style +Parrot 0.8 +X +Parrot 0.6 +X +label +Panda 0.2 +Panda 0.4Input batch +Random +Pairs(Input batch, randperm(Input batch)) +sample pairs +random.choices(Candidates,Weights). +Candidates=[Miaup,CutMia,ResizeMir,Fmia], +Random +Weights = [1,1,1, 1] +mixing method +Mirup +OR +CutMia +OR +ResizeMia +OR +Fmia +Random +wBeta(a,a +w U(0, 1) +入U(0.1,0.8) +wBeta(Q,a +mixing ratio +OR +OR +OR +Mixup +CutMix +ResizeMix +Fmix +Cat (0.52) +Cat (0.21) +Cat (0.15) +Cat (0.39) +Dog (0.48) +Dog (0.79) +Dog (0.85) +Dog (0.61)Fig. 25. Diagram of the label guessing process used in MixMatch, courtesy +[9]. +Distribution alignment tasks are to make the marginal +distribution of predictions on unlabeled data close to the +marginal distribution of ground truth and encourage the +marginal distribution of predictions on unlabeled data +to be close to the marginal distribution of ground truth +labels. Augmentation anchoring feeds multiple strongly +augmented versions of an input into the model and +encourages each output to be close to the prediction for a +weakly-augmented version of the same input. It is shown +in figure 26. +Fig. 26. +Anchoring augmentation. It makes predictions on strong augmen- +tations of the same image (blue) using the forecast for a weakly enhanced +image (green, centre), courtesy [8]. +(xii) FixMatch : +Fixmatch [115] also alleviates the perfor- +mance of semi-supervised learning (SSL), the model is +trained on limited labeled data then the trained model +is used to assign the label to unlabeled data. Fixmatch +first assigned pseudo labels to unlabeled images having a +probability higher than a certain threshold. The model is +forced to make predictions on a strong augmented version +of the unlabeled image to match its prediction with the +pseudo label using cross-entropy loss. (Overall process is +shown in the figure 27) +(xiii) AugMix : +Augmix [49] is a simple and effective data +augmentation that reduces the gap between the distri- +bution of training and test (unseen) data. M operations +are performed with a corresponding random magnitude +of augmentation and at the end, all those images are +merged to produce a new image that widely explores the +semantically equivalent input space around an image. As +shown in the figure below, three operations are performed +separately in three branches and further operations are +also performed for diversity purposes. Finally, all images +Fig. 27. This image shows the procedure of FixMatch, image is taken from +[115]. +are mixed to generate a new image. It is very useful for +robustness. It is shown in figure 28. +Fig. 28. An overall procedure of AugMix augmentation [49], example is from +[49]. +(xiv) Simple Copy-Paste is a Strong Data Augmentation +Method for Instance Segmentation : This method [37] +simply copies and pastes the instances of one image to +another image. It shows promising results and is very +easy to implement. As shown in the figure below, two +images’ instances are pasted to each other on different +scales. It is visually shown in figure 29. +Fig. 29. Image augmentation performed by simple Copy-Paste [37] method, +courtesy [37]. +(xv) Improved Mixed-Example Data Augmentation: These +days state-of-the-art non-label preserving data augmenta- +tion techniques have shown promising results using linear +combinations of the two examples. This paper [120] ex- +plores research questions: i) Why do these methods work? +ii) By proposing new augmentations, is this linearity +important? It is shown in figure 30. + +Classify +. K augmentations. +Average +Sharpen +Unlabeled +ClassifyWeakly- +augmented +Prediction +Pseudo-label +Unlabeled +Model +example +Strongly- +augmented +Prediction +Modelshear_y +Xorig +Wj=0.12 +Xaugmix +trans +Xaug +W2=0.2 +1-m=0.8 +rotate +post +teriz +m=0.2 +equalize +posterizecopy-pasteFig. 30. A visual comparison of linear methods and generalized augmentation +performed by Improved Mixed-Example, image is taken from [120]. +(xvi) RICAP : +Random image cropping and patching (RI- +CAP) [122] is a new data augmentation technique that +cuts and mixes four images rather than two images, +and the labels of the images are also mixed. It shows +impressive performance on popular datasets i.e. CIFAR10 +, CIFAR100, and imageNet. For more detail, RICAP is +shown in the figure 31. +Fig. 31. +A conceptual explanation of the RICAP data augmentation, the +example is from [122]. +(xvii) Rethinking Data Augmentation for Image Super- +resolution: A Comprehensive Analysis and a New +Strategy : +This paper [143] explores and analyses ex- +isting data augmentation techniques for super-resolution +and proposes another data augmentation technique for +super-solution, named cutblur that cuts high-resolution +image patches and pastes to corresponding low-resolution +images and vice-versa. Cutblur shows impressive perfor- +mance on super-resolution. Furthermore, the process is +illustrated in the figure 32 and 33. +Fig. 32. An Schematic illustration of CutBlur operation, image is taken from +[143]. +Fig. 33. A visual comparison between High resolution, low resolution and +CutBlur, courtesy [143]. +(xviii) ResizeMix: Mixing Data with Preserved Object Infor- +mation and True Labels : The ResizeMix [100] method +directly cuts and pastes the source data in 4 different ways +to target the image. 4 different ways including salient part, +non-part, random part or resize source image to patch, as +shown in the figure 34. It addresses two questions: +• How to obtain a patch from the source image? +• where to paste the patch from the source image in the +target image? +Furthermore, it was found that saliency information is +not important to promote mixing data augmentation. +ResizeMix is shown in the figure 34. +(xix) ClassMix: Segmentation-Based Data Augmentation +for Semi-Supervised Learning : This research work [93] +proposed novel data augmentation for semi-supervised +semantic segmentation with inspiration, traditional data +augmentation is not effective for semantic segmentation +as they are for image classification. Proposed data aug- +mentation named ClassMix, which augments the training +sample by mixing unlabeled samples, by exploiting net- +work prediction while considering object boundaries. The +proposed approach showed signification performance on +two common datasets for semi-supervised semantic seg- +mentation. The overall process is shown in the figure 35. +(xx) Context Decoupling Augmentation for Weakly Su- +pervised Semantic Segmentation : +This article [119] + +Input +Linear Methods +GeneralizedCut-and-paste +Cut-and-paste +- +- +- +LR : +HR : +LR +HR +HR +LR +LR +HR (input) +HR +LR (input)HRFig. 34. A visual representation of different cropping manners from the source +image and different pasting manners to the target image, image is taken from +[100]. +Fig. 35. +In a visual representation classMix augmentation, two images are +sampled then based on the predictions of each image a binary mask is created. +The mask is then used to mix the images and their predictions, the image is +taken from [93]. +addresses the problem of traditional data techniques for +WSSS, increasing the same contextual data semantic +samples does not add much value in object differentia- +tion, e.g. image classification, “cat” recognition is due +to the cat itself and also its surrounding context, that +discourages model to focus only on the cat. To break this, +this article proposed a novel data augmentation named +Context Decoupling Augmentation, to make it diverse +where the specific object appears and guide the network +to break the dependencies between object and contextual +information. In this, the way it also provides augmenta- +tion and the network focus to object instance rather than +object instance and contextual information. A comparison +of traditional data augmentation and Context Decoupling +Augmentation is shown below in the figure 36. +(xxi) ObjectAug: Object-level Data Augmentation for Se- +Fig. 36. A visual representation of the difference between the conventional +augmentation approach and context decoupling augmentation (CDA), image +is taken from [119]. +mantic Image Segmentation : +This article [148] ad- +dresses the problem of mixing image-level data augmen- +tation strategies, which failed to operate for segmentation +since at object and background are coupled as bound- +aries of objects are not augmented due to their fixed +semantic bond with the background. To mitigate this +problem, this article proposes a novel approach named +ObjectAug, object-level augmentation for semantic seg- +mentation. First, it separates object(s) and backgrounds +from an image with the help of semantic labels then each +object is augmented using popular data augmentation +techniques such as flipping and rotating. Pixel changes +due to these data augmentations are restored using image +inpainting. In the end, the object(s) and background are +coupled to create an augmented image. Experimental re- +sults suggest that ObjectAug has shown effective perfor- +mance improvement for segmentation tasks. Furthermore, +ObjectAug is shown in the figure 37. +C. Advance Data Augmentation Methods +AutoAugment: The goal of this technique is to find the +data augmentation policies from training data. It solves the +problem of finding the best augmentation policy as a discrete +search problem. It consists of a search algorithm and a search +space. It is divided into two parts. +• Reinforcement learning data augmentation +• Non-Reinforcement learning data augmentation +1) Reinforcement Learning data augmentations: Rein- +forcement learning data augmentaion technique generalize and +improve the performance of deep networks in an environment. +(i) AutoAugment : This work [23] automatically finds the +best data augmentation rather than manual data augmen- +tation. To address this limitation, this article proposes +autoaugment, where search space is designed and has +policies consisting of many sub-policies. Each subpolicy +has two parameters one is image processing function and + +Source patch P +Source image Is +Salient region +Target image It +iNon-salient region +How to obtain? +Where to paste? +Random region +Resizefe Conventional +Augmentation +Rotation +Color jittering +Raw image +CDAFig. 37. +ObjectAug can perform various augmentation methods for each +object to boost the performance of semantic segmentation. The left husky +is scaled and shifted, while the right one is flipped and shifted. Thus, +the boundaries between objects are extensively augmented to boost their +performance, the example is from [148]. +the second one is the probability with magnitude. These +subpolicies are found using reinforcement learning as a +search algorithm. The overall process is shown in the +figure 38. +Fig. 38. +A visual overview of the sub-policies from ImageNet using +AutoAugment, example is from [23]. +(ii) Fast AutoAugment : Fast Autoaugment [82] addresses +the problem of autoaugment, it takes a lot of time to +find the optimal data augmentation strategy. To end this, +fast auto augment finds more optimal data augmentations +using an efficient search strategy based on density match- +ing. It reduces the higher order of training time compared +to auto augment. The overall procedure is shown in +figure 39. +(iii) Faster AutoAugment: This article proposes a faster +autoaugment [46] policy intending to find effective data +augmentation policies very efficiently. Faster autoaug- +ment is based on a differentiable augmentation searching +policy and additionally, it not only estimates gradients +for many transformation operations having discrete pa- +rameters but also provides a mechanism for choosing +operations efficiently. Moreover, it introduces a training +Fig. 39. An overall procedure of augmentation search by Fast AutoAugment +algorithm, courtesy [82]. +objective function with aim of minimising the distance +between original and augmented distribution, that is also +differentiable. Parameters of augmentations are updated +during backpropagation. The Overall process is defined +in figure 40: +Fig. 40. +An Overview of the Faster AutoAugment augmentation, image is +taken from [46]. +(iv) Reinforcement Learning with Augmented Data: This +paper proposes Reinforcement Learning with Augmented +Data (RAD) [76], easily pluggable and enhances the +performance of RL algorithms by targeting two issues +i) learning data efficiency ii) generalisation capability +for new environments. Furthermore, it shows traditional +data augmentation techniques enable RL algorithms to +outperform complex SOTA tasks for pixel-based control +and state-based control. Overall process is defined in + +(a)Original image +(b) Original label +(c) Augmented image +(d) Augmented labelOriginal +Sub-policy 1 +Sub-policy 2 +Sub-policy3 +Sub-policy 4 +Sub-policy5 +Batch 1 +Batch2 +Batch3 +Equalize, 0.4, 4 +Solarize.0.6,3 +Posterize.0.8.5 +Rotate.0.2.3 +Equalize.0.6.8 +Rotate, 0.8,8 +Equalize,0.6,7 +Equalize, 1.0, 2 +Solarize,0.6,8 +Posterize,0.4,6(K) +D(K) +sample B +select N +D(K) +M +train +evaluate +(y)Q +M(0) +train +split ++++ +(1) +sample B +select N +Augment +Policy +D() +M +train +evaluate +apply +D(1) +train +M(0) +T(Dtrain) +trainOriginal or +Classified +Augmented? + +correctly? +Critic +Policy +Update policy by +backpropagationfigure 41: +Fig. 41. An overview of different augmentation investigated in RAD, example +is from [76]. +(v) LOCAL PATCH AUTOAUGMENT WITH MULTI- +AGENT COLLABORATION: +This is the first paper +[83] that finds data augmentation policy for patch level +using reinforcement learning, named multi-agent rein- +forcement learning (MARL). MARL starts by dividing +images into patches and jointly finds optimal data aug- +mentation policy for each patch. It shows competitive +results on SOTA benchmarks. MARL is compared and +differentiated with other augmentation. Overall process +is defined in figure 42: +Fig. 42. An Illustration of different automated augmentation policies, courtesy +[83]. +(vi) Learning Data Augmentation Strategies for Object +Detection: This work [155] proposes to use autoaugment +that learns the best policies for object detection. It finds +the best value and then compares it with the value of +architecture. It addresses two key issues of augmentation +for object detection, +a) Classification learned policies can not directly be ap- +plied for detection tasks, and it adds more complexity +to deal with bounding boxes in a case if geometric +augmentations are applied. +b) Most research thinks it adds much less value compared +to designing new network architecture so gets less +attention but augmentation for object detection should +be selected carefully. +Some sub-policies for this data augmentation are shown +below. +Fig. 43. +Different data augmentation sub-policies explored, image is taken +from, [155]. The sub-policies details are given below. +Sub-policy 1. (Color, 0.2, 8), (Rotate, 0.8, 10) +Sub-policy 2. (BBox Only ShearY, 0.8, 5) +Sub-policy 3. (SolarizeAdd, 0.6, 8), (Brightness, 0.8, 10) +Sub-policy 4. (ShearY, 0.6, 10), (BBox Only Equalize, 0.6,8) +Sub-policy 5. (Equalize, 0.6, 10), (TranslateX, 0.2, 2) +(vii) Scale-aware Automatic Augmentation for Object De- +tection: This work [18] proposes a new data augmenta- +tion for object detection named scale aware autoAug, first, +it defines a search space where image level and box level +data augmentation are prepared for scale invariance, sec- +ondly, this work also proposes a new search metric named +Pareto scale balance for search augmentation effectively +and efficiently. Some examples of data augmentation are +shown in figure 44. +(viii) ADA: Adversarial Data Augmentation for Object +Detection: Data augmentation for object detection has +improved performance but it is difficult to understand +whether these augmentations are optimal or not. This + +Input +Crop +Translate +Window +Grayscale +Cutout +Flip +Rotate +Cutout-color +Random conv +Color-jitter +Augmentations applied +consistentlyacross +stacked framesOriginal +Images +Image-Wise +AutoAugment +Patch-Wise +AutoAugmentBatch 1 +Batch 2 +Batch 3 +Batch 4 +Sub-policy 1 +Sub-policy 2 +Sub-policy 3 +Sub-policy 4 +Sub-policy 5Fig. 44. +Example of scale-aware search space which includes image level +and box-level augmentation, the example is from, [18]. +article [7] provides a systematic way to find optimal +adversarial perturbation of data augmentation from an +object detection perspective, that is based on game- +theoretic interpretation aka Nash equilibrium of data. +Nash equilibrium provides the optimal bounding box pre- +dictor and optimal design for data augmentation. Optimal +adversarial perturbation refers to the worst perturbation of +ground truth, that forces the box predictor to learn from +the most difficult distribution of samples. An example is +shown in figure 45. +Fig. 45. +Annotation distribution types. Adversarial augmentation chooses +bounding boxes that are as distinct from the truth as possible while yet +containing crucial object characteristics. The example is from, [7]. +(ix) Deep CNN Ensemble with Data Augmentation for Ob- +ject Detection: This article [42] proposes a new variant of +the R-CNN model with two core modifications in training +and evaluation. First, it uses several different CNN mod- +els as ensembler in R-CNN, secondly, it smartly augments +PASCAL VOC training examples with Microsoft COCO +data by selecting a subset from Microsoft COCO datasets +that are consistent with PASCAL VOC. Consequently, the +dataset size is enlarged and improves the performance. +The schematic diagram is shown in the figure 46. +(x) Robust and Accurate Object Detection via Adversarial +Learning: This [16] first shows classifier performance +gain from different data augmentations when fine-tuned +to object detection tasks disappears and performance in +terms of accuracy and robustness is not improving. The +article provides a unique way of exploring adversarial +samples that helps to improve performance. To do so, it +augments the example during the fine-tuning stage for +Fig. 46. +The proposed schematic diagram. Example is from, [42]. +object detectors by exploring adversarial samples, which +is considered model-dependent data augmentation. First, +it picks the stronger adversarial sample from detector +classification and localization layers and these change +with the detector to ensure augmentation policy remains +consistent. It showed significant performance gain in +terms of accuracy and robustness on different object +detection tasks. +(xi) Perspective Transformation Data Augmentation for +Object Detection: This article [129] proposes a new +data augmentation for objection detection named perspec- +tive transformation that generates new images captured +at different angles. Thus, it mimics images as if they +are taken at a certain angle where the camera can not +capture those images. This method showed effectiveness +on several object detection datasets. An example of the +proposed data augmentation is shown in the figure below. +(xii) Deep Adversarial Data Augmentation for Extremely +Low Data Regimes: This article [149] addresses the +issue of extremely low data regimes: labeled data is +at a very low. To deal with that problem, it proposes +a deep adversarial data augmentation (DADA), where +data augmentation is formulated as a problem of training +class conditional and supervised GAN. Furthermore, it +also introduces new discriminator loss with aim of fitting +data augmentation were real and augmented samples are +forced to participate equally and be consistent in finding +decision boundaries. +2) Non-Reinforcement +Learning +data +augmentations: +dummy text here +(i) RandAugment: Previous optimal augmentation finding +uses reinforcement or some complex learning strategy +that takes a lot of time to find. RandAugment augmen- +tation [24] removes obstacles of a separate searching +phase, which makes training more complex and conse- + +Image-level Aug +Sample images by Prob. +Box-level Aug +Box-levelAugpolicy contains +Mag is for zooming ratio. +Zoom-in/out-(Prob,Mag) +Color/Geometric-(Prob,Mag,Area) +ColorandGeometricoperations. +Aug types +Prob. +.Mag.Area ratio +Brightness +P1 +M1 +Color +P2 +M2 +Zoom-out +Largeobject +Contrast +P3 +M3 +W +S0n≥0 +e.g. r(Sbox) <1 +Cutout +P4 +M4 +Color +r(Sbox) +Equalize +Ps +Ms +Sharpness +Ps +M6 +Solarize +P, +M7 +H +SolarizeAdd +P: +Mg +Hflip +Pg +Mg +Rotate +P10o +M1o +Original +Shearx +P11 +M11 +Geometric +r(Sbox) +Pori= 1 - Pin - Pout +Middleobject +Sheary +P12 +M12 +W +e.g. r(sbox) > 1 +TranslateX +P13 +M13 +Translatey +P14 +M14 +r(Sbox) is the ratio of aug area and box size. +H +It varies fordifferent scale of objects. +Zoom-in +Smallobject +"o +e.g. r(sbox) > 1 +Scale-awareBox-levelAug +Bounding box +AugmentedareaSingle Ground Truth +RandomAugmentation +AdversarialAugmentationPrediction +Average +VGG-16 +GoogleNet +VGG-16 +GoogleNet +VGG-16 +GoogleNet +PASCAL2012 +PASCAL2007 +PASCAL2012 +coco Filterec +PASCAL2007 +PASCAL2012 +COCO2014Fig. 47. Overview of Robust and Accurate Object detection via adversarial +learning. In the top image, it improves object detector accuracy on clean im- +ages. In middle, improves the detector’s robustness against natural corruption, +and at the bottom, it improves the robustness against cross-dataset domain +shift. The image is taken from, [18]. +quently adds computational cost overhead. To break this, +randaugment random applies N data augmentations with +M magnitude of all augmentations. Some visualisation is +demonstrated in the figure 48: +3) Neural Style Transfer:: It is another category of data +augmentation, which can transfer the artist style of one image +to another without changing semantics at a high level. It brings +more variety to the training set. The main objective of this +neural style transfer is to generate a third image from two +images, where one image provides texture content and another +provides high-level semantic content. +(i) STaDA: Style Transfer as Data Augmentation : This +work [153] thoroughly evaluated different SOTA neural +style transfer algorithms as data augmentation for image +classification tasks. It shows significant performance gain +on Caltech 101 and Caltech 256 datasets. Furthermore, +it also combines neural style transfer algorithms with +conventional data augmentation methods. A sample of +this augmentation is shown in figure 49. +Fig. 48. Example images augmented by RandAugment, image is taken from +[24]. +Fig. 49. Overview of the original image and two stylized images by STaDA. +Image is taken from, [153]. +(ii) Neural Style Transfer as Data Augmentation for +Improving COVID-19 Diagnosis Classification : This +work [51] shows the effectiveness of a cycle generative +adversarial network (GAN), which is mostly used for +neural style transfer, augments COVID-19 negative x-ray +image to convert into positive COVID image to balance +the dataset and also to increase the diversity of the dataset. +It shows that augmenting the images with Cycle GAN +can improve performance over several different CNN +architectures. A sample of this augmentation is shown +in figure 50. +(iii) Style Augmentation: Data Augmentation via Style +Randomization: This work [59] proposed a novel data +augmentation named style augmentation (SA) based on +style neural transfer. SA randomizes the color, contrast, +and texture while maintaining the shape and semantic +content during the training. This is done by picking +an arbitrary style transfer network for randomizing the +style and by getting the target style from multivariate +normal distribution embedding. It improves performance +in three different tasks: classification, regression, and + +Vanilla +Det-AdvProp (ours) +potted plant: 47% +pottedplant:37% +spoon:32% +person:98% +person: 93% +bowl:35% +ovenz +48% +oven: 55% +47% +oven: 70% +person: 49% +knife: 32% +bowl: 41% +90wl:50% +bowl: +spoon:49% +bowl: 67% +COCO (+ 0.3~1.1 mAP) +Accurateon Clean Images +person: 76% +person: 79% +person:40% +oven:39% +oven:37% +bowl:34% +COCO-C(+0.8~3.8mAP) +Robust to Natural Corruption +potted plant: 44% +%06 +potted plant: 51% +cat:93% +PASCALVOC (+0.2~1.3mAP) +RobusttoDomain ShiftMagnitude:9 +Original +Shearx +AutoContrast +Magnitude:17 +Original +Shearx +AutoContrast +Magnitude:28 +Original +Shearx +AutoContrast(a) Original +(b) Snow +(c) YourNameFig. 50. +Overview of generating synthetic covid images from the healthy +category. As the no of epochs grows the quality of the synthetic images +improves. Example is from [51]. +domain adaptation. The style augmentation sample is +shown in figure 51. +Fig. 51. Overview of Style augmentation applied to an image. The shape is +preserved but the style, including color, texture, and contrast is randomized. +Image is from [59]. +(iv) StyPath: Style-Transfer Data Augmentation for Ro- +bust Histology Image Classification: This paper [22] +proposes a novel pipeline for Antibody Mediated Rejec- +tion (AMR) classification in kidneys based on StyPath +data augmentation. StyPath is data augmentation that +transfers style intending to reduce bias. The proposed +augmentation is much faster than SOTA augmentations +for AMR classification. Some samples are shown in +figure 52. +(v) A Neural Algorithm of Artistic Style : This work [36] +introduces an artificial system (AS) based on Deep neural +network that generates artistic images of high perceptual +quality. AS creates neural embedding and then AS uses +the embedding to separate the style and content of the +image and then recombines the content and style of target +images to generate the artistic image. The sample is +shown in figure 53 +4) Feature space data augmentations: Feature data aug- +mentation is another category of data augmentation, where first +images are first transformed into embedding or representation +then data augmentation is performed on the embedding of the +image. Recently a few works have been done in this area, we +selectively highlight the work in a precise way. +Fig. 52. Comparison of content and random initialization. Authors observe +that output images initialized as noise appeared distorded and discolored and +failed to retain the content fidelaty. Image is from [22]. +Fig. 53. Overview of the styled image by neural algorithm. Image is from +[36]. +(i) Dataset Augmentation in Feature Space : This work +[26] first used encoder-decoder to learn representation, +then on representation apply different transformations +such as adding noise, interpolating, or extrapolating. The +proposed method has shown performance improvement +on both static and sequential data. +Fig. 54. +Overview of interpolation and extrapolation between handwritten +characters. Original characters are shown in bold. Image is taken from [26]. +(ii) Feature Space Augmentation for Long-Tailed Data : +This paper [21] proposed the novel data augmentation in +feature space to address the long-tailed issue and uplift +the under-represented class samples. The proposed ap- +proach first separates class-specific features into generic +and specific features with the help of class activation +maps. Under-represented class samples are generated +by injecting class-specific features of under-represented +classes with class-generic features from other confusing + +Epochs +5 +10 +15 +20 +25 +3010 iter +50 iter +100 iter +100 iter +X +cont init +noise_init +X +X +X +sty +ino +inoQ +a +a +8 +8 +a +@ +8 +8 +Q +d +(a) Interpolation +(b) Extrapolationclasses. This enables diverse data and also deals with +the problem of under-represented class samples. It has +shown SOTA performance on different datasets. As it is +demonstrated in figure 55. +Fig. 55. +Left: limited but well-spread data. Right: Without sufficient data. +Image is taken from [21]. +(iii) Adversarial Feature Augmentation for Unsupervised +Domain Adaptation: Generative Adversarial Networks +(GANs) showed promising results in unsupervised do- +main adaptation to learn target domain features indis- +tinguishable from the source domain. This work [127] +extends GAN to force features extractor to be domain- +invariant ii) To train it via data augmentation in feature +space, named feature augmentation. This work explores +data augmentation at the feature level with GAN. +(iv) Understanding data augmentation for classification: +when to warp? : This paper [138] investigates the data +augmentation advantages on image space and feature +space during training. It proposed two approaches i) +data warping which generates extra samples in image +space using data augmentations and ii) synthetic over- +sampling, which generates samples in feature space. It +also suggests that it is possible to apply general data +augmentation techniques in feature space if reasonable +data augmentations for data are known. +(v) FeatMatch: Feature-Based Augmentation for Semi- +Supervised Learning : This work [73] presents a novel +approach of data augmentation in features space for SSL +inspired by an image-based SSL method that uses a com- +bination of augmentations of the images and consistency +regularization. Image-based SSL methods are restricted to +only conventional data augmentation. To break this end, +the feature-based SSL method produced diverse features +from complex data augmentations. One key point is, these +advanced data augmentations exploit the information +from both intra-class and inter-class representations ex- +tracted via clustering. The proposed method only showed +significant performance gain on min-Imagenet such as an +absolute 17.44% gain on miniImageNet, but also showed +robustness on samples that are out-of-distribution. More- +over, the difference between image-level and feature-level +augmentation and consistency is shown in figure 56. +Fig. 56. +An overview of featMatch augmentation applied on images and +features. Image is taken from [21]. +III. RESULTS +In this section, we provide the detailed result for various +CV tasks such as image classification, object detection, and +semantic segmentation. The main purpose is to show the effect +of the data augmentation in CV different tasks and to do so, we +compile results from various SOTA data augmentation works. +A. Image Classification +In this section, we present the result of several SOTA +data augmentation methods for supervised learning and semi- +supervised learning. Both are discussed below: +1) supervised learning results: Supervised learning is, we +have data on a large quantity that is wholly labeled. We train +NN on that data. In this section, we compare and compile the +results from several SOTA data augmentation methods and +put them in two different tables as shown in table II-A3d and +table II. In table II-A3d results, + sign shows traditional data +augmentations such as flipping, rotating and cropping, have +been used along with SOTA augmentation method. The used +datasets are CIFAR10, CIFAR100 and ImageNet, and the used +networks are wideresnet flavours, pyramid network flavours +and several popular resnet flavours. All the classification +results are reported in accuracy, the higher is the best. As +it is noticed from table II-A3d and table II that each data +augmentation has significantly improved the accuracy. +2) Semi-supervised +learning: +Semi-supervised +learning +(SSL) is when we have limited labeled data but unlabeled +data is available on large scale. Labeling the unlabeled data +is tedious, time-consuming and cost [71], [139]. To avoid +these issues, SSL is used. There are several techniques of +SSL, but recently data augmentation is employed with the +limited labeled data to increase the diversity of the data. +Data augmentation with SSL has increased the performance +on different datasets and NN architectures. The used dataset +are CIFAR10, CIFAR100, SVHn and Mini-ImageNet. Several + +Learned +Boundary +Learned +Boundary +True +True +Boundary +Boundary +Re-weight +Samplefc l + softmax +x +Encoder +layer +fx +p(ylf) +Image-based +Consistency Loss +Augmentation +fc layer + softmax +x +Encoder +fx +p(ylfx) +(a) Image-Based Augmentation and Consistency +fc layer + softmax +gx +Prototypes +per class ★ +p(ylgx) +Feature-based +Consistency Loss +Augmentation +fc l + softmax +Encoder +layer +x +fx +p(ylfx) +(b) Feature-Based Augmentation and ConsistencySSL techniques are used. We compile the results from many +SOTA SSL methods with data augmentation and present in +this work. The effect of the data augmentation has also been +shown with the different number of samples in SSL as shown +in table III, table IV and table V. +B. Object detection +In this section, we discuss the effectiveness of various +image data augmentation techniques on the frequently used +COCO2017, PASCAL VOC, VOC 2007, and VOC 2012 +datasets, which are commonly used for object detection tasks. +We compile results from various SOTA data augmentation +methods and put them in three different tables as shown in +the table III-B, VII, and VIII. FRCN along with synthetic +data gives the best mAP accuracy on VOC 2007 dataset as +shown in Table VII. Several classical and automatic data +augmentation methods have shown the promising performance +on different state-of-the-art models on PASCAL VOC dataset +as shown in table III-B. The DetAdvProp achieves the highest +score on every model and mAP, AP50 and AP75 metrics +on PASCAL VOC 2012 dataset, outperforming AutoAugment +[23] as shown in the table VIII. The performance is reported +in average precision (AP). AP50 and AP75 are the average +precision with 50% and 75% threshold, respectively. +C. Semantic Segmentation +This subsection includes semantic segmentation results on +PASCAL VOC and CITYSCAPES datasets, most frequently +used in several research papers. In the table (IX) and table +(X), we compiled the effectiveness of validation set results +on the different datasets (mIoU) with data augmentation on +semantic segmentation models; the best results of performance +(mIoU) accuracy on the Cityscape dataset as shown in ta- +ble (ix) and best results of performance (mIoU) accuracy +on Pascal VOC datasets are shown in table (x). We found +performance gains on a few metrics with several semantic +segmentation models: deeplabv3+ [144], DeepLab-v2 [93], +Xception-65 [144], ExFuse [150] and Eff-L2 [156] . All +semantic segmentation models have been found to perform +better when data augmentation techniques are used. Traditional +data augmentation methods are rotation, scaling, flipping and +shifting [148]. +IV. DISCUSSION AND FUTURE DIRECTIONS +A. Current approaches +It is proven that if we provide more data to the model, +it improves model performance +[43], [121]. A few current +tendencies are discussed by Xu et al. [141]. Among these, one +way is to collect the data and label it manually, but it is not an +efficient way to do this. Another efficient way is to apply data +augmentation, the more data augmentations we apply, the more +performance improves to a certain extent. Currently, image +mixing methods and autoaugment methods are successful for +image classification tasks, scale aware based auto augment +methods are showing promising results in detection tasks and +semantic segmentation tasks. But these data augmentation +performances can vary with the number of data augmentation +applied, as it is known that the combined data augmentation +methods show better performance than single one [97], [142]. +B. Theoretical aspects +There is no theoretical support available to explain why +specific augmentation is improving performance and which +sample(s) should be augmented, as the same aspect has been +discussed by Yang et al [142] and Shorten et al [108]. Like +in random erasing, we randomly erase the region of the +image - sometime may erase discriminating features, and the +erased image makes no sense to a human. But the reason +behind performance improvement is still unknown, which is +another open challenge. Most of the time, we find the optimal +parameters of the augmentation through an extensive number +of experiments or we choose data augmentation based on our +experience. But there should be a mechanism for choosing the +data augmentation with theoretical support considering model +architecture and dataset size. Researching the theoretical as- +pect is another challenge open for the research community. +C. Optimal number of samples generation +It is a known fact, as we increase data size, it improves +the performance +[43], [108], [121], [142] but it is not a +case - increasing the number of samples will not improve +performance after a certain number of samples +[70]. What +is the optimal number of samples to be generated, depending +on the model architecture and dataset size, is another aspect to +be explored. Currently, researchers perform many experiments +to find the optimal number of sample generation [70]. But it +is not feasible way as it requires time and computational cost. +Can we devise a mechanism to find an optimal number of +samples, which is an open research challenge? +D. Selection of data augmentation based on model archi- +tecture and dataset +Data augmentation selection depends on the nature of the +dataset and model architecture. Like on MNIST [25] dataset, +geometric transformations are not safe such as rotation on 6 +and 9 digits will no longer preserve the label information. For +densely parameterized CNN, it is easy to overfit on weakly +augmented datasets, and for shallow parameterized CNN, it +may break generalization capability with data augmentation. +It suggests, while selecting the data augmentation, the nature +of the dataset and model architecture should be taken into +account. It is not an easy problem to solve. Currently, numer- +ous experiments are performed to find model architecture and +suitable data augmentation for a specific dataset. Devising a +systematic approach to select the data augmentation based on +dataset and model architecture is another open challenge. +E. Augmentations for spaces +Most of the data augmentation has been explored on the +image level - data space. Very few research works have +explored data on feature level - feature space. Challenges here +arise, in which space should we apply data augmentation, data + +TABLE III +COMPARISON ON CIFAR-10 AND SVHN. NUMBER REPRESENTS ERROR RATES ACROSS THREE RUNS. +CIFAR-10 +SVHN +Method +40 labels +250 labels +1,000 labels +4,000 labels +40 labels +250 labels +1,000 labels +4,000 labels +VAT [91] +- +36.03 ± 2.82 +18.64 ± 0.40 +11.05 ± 0.31 +- +8.41 ± 1.01 +5.98 ± 0.21 +4.20 ± 0.15 +Mean Teacher [123] +- +47.32 ± 4.71 +17.32±4.00 +10.36±0.25 +- +6.45±2.43 +3.75±.10 +3.39±0.11 +MixMatch [9] +47.54±11.50 +11.08±.87 +7.75±.32 +6.24±.06 +42.55±14.53 +3.78±.26 +3.27±.31 +2.89±.06 +ReMixMatch [8] +19.10±9.64 +6.27±0.34 +5.73±0.16 +5.14±0.04 +3.34±0.20 +3.10±0.50 +2.83±0.30 +2.42±0.09 +UDA +29.05±5.93 +8.76± 0.90 +5.87± 0.13 +5.29± 0.25 +52.63±20.51 +2.76± 0.17 +2.55± 0.09 +2.47± 0.15 +SSL with Memory [17] +- +- +- +11.9±0.22 +- +8.83 +4.21 +- +Deep Co-Training [99] +- +- +- +8.35± 0.06 +- +- +3.29 ±0.03 +- +Weight Averaging [5] +- +- +15.58 I 0.12 +9.05± 0.21 +- +- +- +- +ICT [126] +- +- +15.48 I 0.78 +7.29± 0.02 +- +4.78 I 0.68 +3.89 ±0.04 +- +Label Propagation [57] +- +- +16.93 ± 0.70 +10.61 ± 0.28 +- +- +- +- +SNTG [87] +- +- +18.41 ± 0.52 +9.89 ±0.34 +- +4.29± 0.23 +3.86 ±0.27 +- +PLCB [4] +- +- +6.85 ±0.15 +5.97± 0.15 +- +- +- +- +II-model [105] +- +53.02 ±2.05 +31.53 ± 0.98 +17.41± 0.37 +- +17.65 ±0.27 +8.60± 0.18 +5.57± 0.14 +PseudoLabel [77] +- +49.98 ±1.17 +30.91 ±1.73 +16.21 ± 0.11 +- +21.16± 0.88 +10.19 ± 0.41 +5.71± 0.07 +Mixup [147] +- +47.43 ± 0.92 +25.72 ± 0.66 +13.15 ± 0.20 +- +39.97 ± 1.89 +16.79 ± 0.63 +7.96 ±0.14 +FeatMatch [73] +- +7.50 ±0.64 +5.76 ±0.07 +4.91± 0.18 +- +3.34± 0.19 +3.10± 0.06 +2.62 ±0.08 +FixMatch [115] +13.81±3.37 +5.07±0.65 +- +4.26±0.05 +3.96±2.17 +2.48±0.38 +2.28±0.11 +- +SelfMatch [62] +93.19±1.08 +95.13±0.26 +- +95.94±0.08 +96.58±1.02 +97.37±0.43 +97.49±0.07 +- +TABLE IV +COMPARISON ON CIFAR-100 AND MINI-IMAGENET. NUMBER REPRESENTS ERROR RATES ACROSS TWO RUNS. +CIFAR-100 +mini-ImageNet +Method +400 labels +4,000 labels +10,000 labels +4,000 labels +10,000 labels +II-model [105] +- +- +39.19± 0.36 +- +- +SNTG [87] +- +- +37.97± 0.29 +- +- +SSL with Memory [17] +- +- +34.51± 0.61 +- +- +Deep Co-Training [99] +- +- +34.63± 0.14 +- +- +Weight Averaging [5] +- +- +33.62± 0.54 +- +- +Mean Teacher [123] +- +45.36 ±0.49 +36.08± 0.51 +72.51± 0.22 +57.55 ± 1.11 +Label Propagation [57] +- +43.73 ±0.20 +35.92 ±0.47 +70.29± 0.81 +57.58 ±1.47 +PLCB [4] +- +37.55 ±1.09 +32.15 ±0.50 +56.49 ±0.51 +46.08 ± 0.11 +FeatMatch +- +31.06 ± 0.41 +26.83 ± 0.04 +39.05 0.06 +34.79±0.22 +MixMatch +67.61±1.32 +- +28.31±0.33 +- +- +UDA +59.28±0.88 +- +24.50±0.25 +- +- +ReMixMatch +44.28±2.06 +- +23.03±0.56 +- +- +FixMatch +48.85±1.75 +- +22.60±0.12 +- +- +space or feature space? It is another interesting aspect that can +be explored. For approaches, it seems it depends on the dataset, +model architecture and task. Mixing augmentations in feature +space is senseless. Currently, approaches are conducting ex- +periments in data space and feature space and then selecting +the best one [138]. This is not the optimal way to go. It is still +an open challenge to be solved. +F. Open research questions +Despite the success of data augmentation techniques in dif- +ferent CV tasks, it still failed to solve challenges in SOTA data +augmentation techniques. After thoroughly reviewing SOTA +data augmentation approaches, we found several challenges +and difficulties, which are yet to be solved, as it is listed below: +• In image mixing techniques, label smoothing has been +used. It makes sense whatever portion of images is mixed, +corresponding labels should be mixed accordingly. To the +best of our knowledge, none has explored label smoothing +for image manipulation and image erasing subcategories. +For example, if the image portion is randomly cut out in +cutout data augmentation, the corresponding label should +be smoothened. The same rule applies to the image +erasing category and image manipulation - where the +image part is lost. +• Currently, data augmentation is performed without con- +sidering the importance of an example. All examples may +not be difficult for the neural network to learn, but some +are. So augmentation should be applied to those difficult +examples by measuring the importance of the examples. +• In image mixing data augmentations, if we mix more +than two images salient parts, that are truly participating +in augmentation unlike RICAP [122], what is its effect? +Note, the corresponding labels of these images will be +mixed accordingly. +• In random data augmentation under auto augmentations, +the order of augmentations has not been explored. We be- +lieve it has significant importance. What are the possible +ways to explore the order of existing augmentations such + +TABLE V +COMPARISON OF TEST ERROR RATES ON CIFAR-10 & SVHN USING WIDERESNET-28 AND CNN-13. +Approach +Method +CIFAR-10 (Nl=4000) +SVHN(Nl=1000) +WideResNet-28 +Supervised +20.26 ± 0.38 +12.83 ± 0.47 +Pseudo +PL [77] +17.78 ± 0.57 +7.62 ± 0.29 +Labeling +PL-CB [4] +6.28 ± 0.3 +- +II Model [75] +16.37 ± 0.63 +7.19 ± 0.27 +Mean Teacher [123] +15.87 ± 0.28 +5.65 + 0.47 +VAT [91] +13.86 ± 0.27 +5.63 ± 0.20 +Consistency +VAT + EntMin [91] +13.13 I 0.39 +5.35 + 0.19 +LGA + VAT [58] +12.06 ± 0.19 +6.58 ± 0.36 +Regularization +ICT [126] +7.66 ± 0.17 +3.53 ± 0.07 +MixMatch [9] +6.24 ± 0.06 +3.27 ± 0.31 +UDA +5.29 ± 0.25 +2.46 ± 0.17 +ReMixMatch (Berthelot et al. 2020) +5.14 ±0.04 +2.42 ± 0.09 +FixMatch [115] +4.26 ± 0.05 +2.28 ± 0.11 +CL +8.92 ± 0.03 +5.65 ± 0.11 +Pseudo +CL+FA [82] +5.51 0.14 +2.90 ± 0.19 +Labeling +CL+FA [82]+Mixup [147] +5.09 ± 0.18 +2.75 ± 0.15 +CL+RA+Mixup [147] +5.27 ± 0.16 +2.80 ± 0.188 +CNN-13 +Pseudo Labeling +TSSDL-MT +9.30 ± 0.55 +3.35 ± 0.27 +LP-MT +10.61±0.28 +- +Ladder net [102] +12.36±0.31 +- +MeanTeacher [123] +12.31 ± 0.24 +3.95 ± 0.19 +Temporal ensembling [75] +12.16 ± 0.24 +4.42 ± 0.16 +Consistency +VAT [91] +11.36 ± 0.34 +5.42 +Regularization +NATEntMin [91] +10.55 ± 0.05 +3.86 +SNTG [87] +10.93 ± 0.14 +3.86 ± 0.27 +ICT [126] +7.29 ± 0.02 +2.89 ± 0.04 +Pseudo +CL +9.81 ± 0.22 +4.75 ± 0.28 +Labeling +CL+RA +5.92 ± 0.07 +3.96 ± 0.10 +as first traditional data augmentations and then image +mixing or weight-based? +• If we mix the masks of the objects in data augmentation +for semantic segmentation, How does the model behave +and what is its effect? +• Finding the optimal ordered number of data augmentation +and the optimal number of samples to be augmented +is another open challenge. For example, in randAug +method there are N optimal number of augmentations +was found but it is not known how many samples should +be augmented. +V. CONCLUSION +This survey presents numerous SOTA data augmentation +methods to cope with overfitting problems in computer vision +tasks due to data limitations. We provided a comprehensive +survey for data augmentation, in which we presented novel +taxonomy of advanced data augmentation approaches, an +overview of each SOTA data augmentation, and results of +numerous computer vision tasks such as image classifica- +tion, object detection and semantic segmentation, with data +augmentation effect. We not only compiled the results for +supervised learning tasks but also compiled results for semi- +supervised learning. For result reproducibility, we compiled +the available codes of the data augmentation by following +the proposed taxonomy. We discuss a different aspects of the +data augmentation with its difficulties. Finally, we discuss the +open research questions, which are very promising and open +new doors, and ignite interest in the research community. We +believe that the survey benefits the researchers as follows: i) +Understanding of the data augmentation ii) No need to find the +results for comparison purposes iii) Results can be reproduced +with available codes. +ACKNOWLEDGMENT +This publication has emanated from research [conducted +with the financial support of supported in part by a grant from] +Science Foundation Ireland under Grant number 18/CRT/6223 +and is supported by the ADAPT Centre for Digital Content +Technology which is funded under the SFI Research Cen- +tres Programme (Grant 13/RC/2106/P2), Lero SFI Centre for +Software (Grant 13/RC/2094/P2) and is co-funded under the +European Regional Development Fund. For the purpose of +Open Access, the author has applied a CC BY public copyright +licence to any Author Accepted Manuscript version arising +from this submission +REFERENCES +[1] Jiwoon Ahn, Sunghyun Cho, and Suha Kwak. +Weakly supervised +learning of instance segmentation with inter-pixel relations. In Pro- +ceedings of the IEEE/CVF conference on computer vision and pattern +recognition, pages 2209–2218, 2019. + +Method +Detector +BackBone +AP +AP50 +AP75 +APs +APm +APl +Hand-crafted: +Dropblock [38] +RetinaNet +ResNet-50 +38.4 +56.4 +41.2 +− +− +− +AutoAugment+color Ops [155] +RetinaNet +ResNet-50 +37.5 +- +- +− +− +− +geometric Ops [155] +RetinaNet +ResNet-50 +38.6 +- +- +− +− +− +bbox-only Ops [155] +RetinaNet +ResNet-50 +39.0 +- +- +− +− +− +Mix-up [151] +Faster R-CNN +ResNet-101 +41.1 +- +- +- +- +- +PSIS* [128] +Faster R-CNN +ResNet-101 +40.2 +61.1 +44.2 +22.3 +45.7 +51.6 +Stitcher [19] +Faster R-CNN +ResNet-101 +42.1 +- +- +26.9 +45.5 +54.1 +GridMask [15] +Faster R-CNN +ResNeXt-101 +42.6 +65.0 +46.5 +- +- +- +InstaBoost* [32] +Mask R-CNN +ResNet-101 +43.0 +64.3 +47.2 +24.8 +45.9 +54.6 +SNIP (MS test)* [112] +Faster R-CNN +ResNet-101-DCN-C4 +44.4 +66.2 +49.9 +27.3 +47.4 +56.9 +SNIPER (MS test)* [113] +Faster R-CNN +ResNet-101-DCN-C4 +46.1 +67.0 +51.6 +29.6 +48.9 +58.1 +Traditional Aug [142] +Faster R-CNN +ResNet-101 +36.80 +58.0 +40.0 +- +- +- +Traditional Aug* [29] +CenterNet +ResNet-101 +41.15 +58.01 +45.30 +- +- +- +Traditional Aug+ [15] +Faster-RCNN +50-FPN (2×) +37.4 +58.7 +40.5 +- +- +- +Traditional Aug+ [15] +Faster-RCNN +50-FPN (2×)+GridMask (p = 0.3) +38.2 +60.0 +41.4 +- +- +- +Traditional Aug+ [15] +Faster-RCNN +50-FPN (2×)+ GridMask (p = 0.5) +38.1 +60.1 +41.2 +- +- +- +Traditional Aug+ [15] +Faster-RCNN +50-FPN (2×)+ GridMask (p = 0.7) +38.3 +60.4 +41.7 +- +- +- +Traditional Aug+ [15] +Faster-RCNN +50-FPN (2×)+ GridMask (p = 0.9) +38.0 +60.1 +41.2 +- +- +- +Traditional Aug+ [15] +Faster-RCNN +50-FPN (4×) +35.7 +56.0 +38.3 +- +- +- +Traditional Aug+ [15] +Faster-RCNN +50-FPN (4×)+ GridMask (p = 0.7) +39.2 +60.8 +42.2 +- +- +- +Traditional Aug+ [15] +Faster-RCNN +X101-FPN (1×)) +41.2 +63.3 +44.8 +- +- +- +Traditional Aug+ [15] +Faster-RCNN +X101-FPN (2×)) +40.4 +62.2 +43.8 +- +- +- +Traditional Aug+ [15] +Faster-RCNN +X101-FPN (2×)+ GridMask (p = 0.7)) +42.6 +65.0 +46.5 +- +- +- +Traditional Aug+ [15] +Faster-RCNN +X101-FPN (2×)+ GridMask (p = 0.7)) +42.6 +65.0 +46.5 +- +- +- +KeepAugment: [41] +Faster R-CNN +ResNet50-C4 +39.5 +− +− +− +− +− +KeepAugment: [41] +Faster R-CNN +ResNet50-FPN +40.7 +− +− +− +− +− +KeepAugment: [41] +RetinaNet +ResNet50-FPN +39.1 +− +− +− +− +− +KeepAugment: [41] +Faster R-CNN +ResNet101-C4 +42.2 +− +− +− +− +− +KeepAugment: [41] +Faster R-CNN +ResNet101-FPN +42.9 +− +− +− +− +− +KeepAugment: [41] +RetinaNet +ResNet101-FPN +41.2 +− +− +− +− +− +DADAAugment: [80] +RetinaNet +ResNet-50 +35.9 +55.8 +38.4 +19.9 +38.8 +45.0 +DADAAugment: [80] +RetinaNet +ResNet-50(DADA) +36.6 +56.8 +39.2 +20.2 +39.7 +46.0 +DADAAugment: [80] +Faster R-CNN +ResNet-50 +36.6 +58.8 +39.6 +21.6 +39.8 +45.0 +DADAAugment: [80] +Faster R-CNN +ResNet-50 (DADA) +37.2 +59.1 +40.2 +22.2 +40.2 +45.7 +DADAAugment: [80] +Mask R-CNN +ResNet-50 +37.4 +59.3 +40.7 +22.2 +40.6 +46.3 +DADAAugment: [80] +Mask R-CNN +ResNet-50(DADA) +37.8 +59.6 +41.1 +22.4 +40.9 +46.6 +AutoAugment: [16] +EfficientDet D0 +EfficientNet B0 +34.4 +52.8 +36.7 +53.1 +40.2 +13.9 +Det-AdvProp: [16] +EfficientDet D0 +EfficientNet B0 +34.7 +52.9 +37.2 +54.1 +40.6 +13.9 +AutoAugment: [16] +EfficientDet D1 +EfficientNet B1 +40.1 +59.2 +43.2 +57.9 +45.7 +19.9 +Det-AdvProp: [16] +EfficientDet D1 +EfficientNet B1 +40.5 +59.2 +43.3 +58.8 +46.2 +20.6 +AutoAugment: [16] +EfficientDet D2 +EfficientNet B2 +43.5 +62.8 +46.6 +59.8 +48.7 +23.9 +Det-AdvProp: [16] +EfficientDet D2 +EfficientNet B2 +43.8 +62.6 +47.3 +61.0 +49.6 +25.6 +AutoAugment: [16] +EfficientDet D3 +EfficientNet B3 +47.0 +66.0 +50.8 +63.0 +51.7 +29.8 +Det-AdvProp: [16] +EfficientDet D3 +EfficientNet B3 +47.6 +66.3 +51.4 +64.0 +52.2 +30.2 +AutoAugment: [16] +EfficientDet D4 +EfficientNet B4 +49.5 +68.7 +53.7 +64.9 +54.0 +31.9 +Det-AdvProp: [16] +EfficientDet D4 +EfficientNet B4 +49.8 +68.6 +54.2 +65.2 +54.2 +32.4 +AutoAugment: [16] +EfficientDet D5 +EfficientNet B5 +51.5 +70.4 +56.0 +65.2 +56.1 +35.4 +Det-AdvProp: [16] +EfficientDet D5 +EfficientNet B5 +51.8 +70.7 +56.3 +66.1 +56.2 +36.2 +Automatic: +AutoAug-det [155] +RetinaNet +ResNet-50 +39.0 +- +- +- +- +- +AutoAug-det [155] +RetinaNet +ResNet-101 +40.4 +- +- +- +- +- +AutoAugment [23] +RetinaNet +ResNet-200 +42.1 +- +- +- +- +- +AutoAug-det’ [155] +RetinaNet +ResNet-50 +40.3 +60.0 +43.0 +23.6 +43.9 +53.8 +RandAugmnet* [24] +RetinaNet +ResNet-200 +41.9 +- +- +- +- +- +AutoAug-det [155] +RetinaNet +ResNet-101 +41.8 +61.5 +44.8 +24.4 +45.9 +55.9 +RandAug [24] +RetinaNet +ResNet-101 +40.1 +- +- +- +- +- +RandAug? [10] +RetinaNet +ResNet-101 +41.4 +61.4 +44.5 +25.0 +45.4 +54.2 +Scale-aware AutoAug [18] +RetinaNet +ResNet-50 +41.3 +61.0 +441 +25.2 +44.5 +54.6 +Scale-aware AutoAug +RetinaNet +ResNet-101 +43.1 +62.8 +46.0 +26.2 +46.8 +56.7 +Scale-aware AutoAug +Faster R-CNN +ResNet-101 +44.2 +65.6 +48.6 +29.4 +47.9 +56.7 +Scale-aware AutoAug (MS test) +Faster R-CNN +ResNet-101-DCN-C4 +47.0 +68.6 +52.1 +32.3 +49.3 +60.4 +Scale-aware AutoAug +FCOS +ResNet-101 +44.0 +62.7 +47.3 +28.2 +47.8 +56.1 +Scale-aware AutoAug +FCOS +ResNeXt-32x8d-101-DCN +48.5 +67.2 +52.8 +31.5 +51.9 +63.0 +Scale-aware AutoAug (1200 size) +FCOS +ResNeXt-32x8d-101-DCN +49.6 +68.5 +54.1 +35.7 +52.5 +62.4 +Scale-aware AutoAug (MS Test) +ResNeXt-32x8d-101-DCN +FCOS +51.4 +69.6 +57.0 +37.4 +54.2 +65.1 +TABLE VI +DATA AUGMENTATION EFFECT ON DIFFERENT OBJECT DETECTION METHODS USING PASCAL VOC DATASET + +Method +TSet +mAP +aero +bike +bird +boat +bottle +bus +car +cat +chair +cow +table +dog +horse +mbike +person +plant +sheep +sofa +train +tv +FRCN [39] +7 +66.9 +74.5 +78.3 +69.2 +53.2 +36.6 +77.3 +78.2 +82.0 +40.7 +72.7 +67.9 +79.6 +79.2 +73.0 +69.0 +30.1 +65.4 +70.2 +75.8 +65.8 +FRCN* [132] +7 +69.1 +75.4 +80.8 +67.3 +59.9 +37.6 +81.9 +80.0 +84.5 +50.0 +77.1 +68.2 +81.0 +82.5 +74.3 +69.9 +28.4 +71.1 +70.2 +75.8 +66.6 +ASDN [132] +7 +71.0 +74.4 +81.3 +67.6 +57.0 +46.6 +81.0 +79.3 +86.0 +52.9 +75.9 +73.7 +82.6 +83.2 +77.7 +72.7 +37.4 +66.3 +71.2 +78.2 +74.3 +IRE +7 +70.5 +75.9 +78.9 +69.0 +57.7 +46.4 +81.7 +79.5 +82.9 +49.3 +76.9 +67.9 +81.5 +83.3 +76.7 +73.2 +40.7 +72.8 +66.9 +75.4 +74.2 +ORE +7 +71.0 +75.1 +79.8 +69.7 +60.8 +46.0 +80.4 +79.0 +83.8 +51.6 +76.2 +67.8 +81.2 +83.7 +76.8 +73.8 +43.1 +70.8 +67.4 +78.3 +75.6 +I+ORE +7 +71.5 +76.1 +81.6 +69.5 +60.1 +45.6 +82.2 +79.2 +84.5 +52.5 +78.7 +71.6 +80.4 +83.3 +76.7 +73.9 +39.4 +68.9 +69.8 +79.2 +77.4 +FRCN [39] +7+12 +70.0 +77.0 +78.1 +69.3 +59.4 +38.3 +81.6 +78.6 +86.7 +42.8 +78.8 +68.9 +84.7 +82.0 +76.6 +69.9 +31.8 +70.1 +74.8 +80.4 +70.4 +FRCN* [132] +7+12 +74.8 +78.5 +81.0 +74.7 +67.9 +53.4 +85.6 +84.4 +86.2 +57.4 +80.1 +72.2 +85.2 +84.2 +77.6 +76.1 +45.3 +75.7 +72.3 +81.8 +77.3 +IRE +7+12 +75.6 +79.0 +84.1 +76.3 +66.9 +52.7 +84.5 +84.4 +88.7 +58.0 +82.9 +71.1 +84.8 +84.4 +78.6 +76.7 +45.5 +77.1 +76.3 +82.5 +76.8 +ORE +7+12 +75.8 +79.4 +81.6 +75.6 +66.5 +52.7 +85.5 +84.7 +88.3 +58.7 +82.9 +72.8 +85.0 +84.3 +79.3 +76.3 +46.3 +76.3 +74.9 +86.0 +78.2 +I+ORE +7+12 +76.2 +79.6 +82.5 +75.7 +70.5 +55.1 +85.2 +84.4 +88.4 +58.6 +82.6 +73.9 +84.2 +84.7 +78.8 +76.3 +46.7 +77.9 +75.9 +83.3 +79.3 +SSD +7+12 +77.4 +81.7 +85.4 +75.7 +69.6 +49.9 +84.9 +85.8 +87.4 +61.5 +82.3 +79.2 +86.6 +87.1 +84.7 +78.9 +50.0 +77.4 +79.1 +86.2 +76.3 +SSD+ SD (1x) [129] +7+12 +78.1 +83.2 +84.5 +76.1 +72.1 +50.2 +85.2 +86.3 +87.8 +63.7 +82.8 +80.1 +85.2 +87.2 +84.8 +80.0 +51.5 +77.0 +82.0 +86.1 +76.9 +SSD + SD(2x) [129] +7+12 +78.3 +83.6 +85.0 +76.2 +72.0 +51.3 +85.1 +87.2 +87.6 +64.2 +82.5 +81.9 +85.5 +86.5 +85.9 +81.2 +51.2 +72.3 +82.8 +86.9 +78.4 +SSD +SD(3x) [129] +7+12 +77.8 +80.4 +85.0 +76.3 +70.1 +50.4 +84.8 +86.3 +88.2 +61.0 +83.5 +79.5 +87.2 +86.9 +85.9 +78.8 +51.2 +76.9 +79.4 +86.5 +77.9 +FRCN [39] +7+12 +73.2 +76.5 +79.0 +70.9 +65.5 +52.1 +83.1 +84.7 +86.4 +52.0 +81.9 +65.7 +84.8 +84.6 +77.5 +76.7 +38.8 +73.6 +73.9 +83.0 +72.6 +FRCN+SD(1x) [140] +7 +79.9 +85.1 +86.6 +78.6 +75.7 +65.2 +83.5 +88.4 +88.9 +65.8 +83.6 +74.3 +86.4 +84.7 +85.5 +88.0 +62.0 +75.5 +75.3 +87.7 +76.3 +TABLE VII +VOC 2007 TEST DETECTION AVERAGE PRECISION (%). FRCN* REFERS TO FRCN WITH TRAINING SCHEDULE IN [132] AND SD REFERS TO SYNTHETIC DATA +. + +Model +mAP +AP50 +AP75 +EfficientDet-D0 +55.6 +77.6 +61.4 ++ AutoAugment +55.7 (+0.1) +77.7 (+0.1) +61.8 (+0.4) ++ Det-AdvProp +55.9 (+0.3) +77.9 (+0.3) +62.0 (+0.6) +EfficientDet-D1 +60.8 +82.0 +66.7 ++ AutoAugment +61.0 (+0.2) +82.2 (+0.2) +67.2 (+0.5) ++ Det-AdvProp +61.2 (+0.4) +82.3 (+0.3) +67.4 (+0.7) +EfficientDet-D2 +63.3 +83.6 +69.3 ++ AutoAugment +62.7 (-0.6) +83.3 (-0.3) +69.2 (-0.1) ++ Det-AdvProp +63.5 (+0.2) +83.8 (+0.2) +69.7 (+0.4) +EfficientDet-D3 +65.7 +85.3 +71.8 ++ AutoAugment +65.2 (-0.5) +85.1 (-0.2) +71.3 (-0.5) ++ Det-AdvProp +66.2 (+0.5) +85.9 (+0.6) +72.5 (+0.7) +EfficientDet-D4 +67.0 +86.0 +73.0 ++ AutoAugment +67.0 (+0.0) +86.3 (+0.3) +73.5 (+0.5) ++ Det-AdvProp +67.5 (+0.5) +86.6 (+0.6) +74.0 (+1.0) +EfficientDet-D5 +67.4 +86.9 +73.8 ++ AutoAugment +67.6 (+0.2) +87.2 (+0.3) +74.2 (+0.4) ++ Det-AdvProp +68.2 (+0.8) +87.6 (+0.7) +74.7 (+0.9) +TABLE VIII +RESULTS ON PASCAL VOC 2012. 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Style augmentation: data augmentation + +Method +Model +1/100 +1/50 +1/20 +1/8 +1/4 +Full +GANSeg [116] +VGG16 +- +- +- +- +64.1 +AdvSemSeg [55] +ResNet-101 +- +- +- +- +68.4 +CCT [94] +ResNet-50 +- +- +- +- +69.4 +PseudoSeg [157] +ResNet-101 +- +- +- +- +73.2 +DSBN [144] +ResNet-101 +- +- +- +- +75.0 +DSBN [144] +Xception-65 +- +- +- +- +79.3 +Fully supervised [144] +ResNet-101 +- +- +- +- +78.3 +Fully supervised [144] +Xception-65 +- +- +- +- +79.2 +Adversarial [55] +DeepLab-v2 +- +57.2 +64.7 +69.5 +72.1 +- +s4GAN [90] +DeepLab-v2 +- +63.3 +67.2 +71.4 +- +75.6 +French et.el [35] +DeepLab-v2 +53.79 +64.81 +66.48 +67.60 +- +- +DST-CBC [34] +DeepLab-v2 +61.6 +65.5 +69.3 +70.7 +71.8 +- +ClassMix:Seg* [93] +DeepLab-v2 +54.18 +66.15 +67.77 +71.00 +72.45 +- +Mixup [147] +IRNet +- +- +- +- +- +49 +CutOut [27] +IRNet +- +- +- +- +- +48.9 +CutMix [146] +IRNet +- +- +- +- +- +49.2 +Random pasting [119] +IRNet +- +- +- +- +- +49.8 +CCNN [96] +VGG16 +- +- +- +- +- +35.6 +SEC [66] +VGG16 +- +- +- +- +- +51.1 +STC [135] +VGG16 +- +- +- +- +- +51.2 +AdvEra [134] +VGG16 +- +- +- +- +- +55.7 +DCSP [13] +ResNet101 +- +- +- +- +- +61.9 +MDC [136] +VGG16 +- +- +- +- +- +60.8 +MCOF [131] +ResNet101 +- +- +- +- +- +61.2 +DSRG [54] +ResNet101 +- +- +- +- +- +63.2 +AffinityNet [2] +ResNet-38 +- +- +- +- +- +63.7 +IRNet [1] +ResNet50 +- +- +- +- +- +64.8 +FickleNet [78] +ResNet101 +- +- +- +- +- +65.3 +SEAM [133] +ResNet38 +- +- +- +- +- +65.7 +ICD [31] +ResNet101 +- +- +- +- +- +64.3 +IRNet + CDA [119] +ResNet50 +- +- +- +- +- +66.4 +SEAM + CDA [119] +ResNet38 +- +- +- +- +- +66.8 +DeepLab V3 [148] +MobileNet +- +- +- +- +- +71.9 +DeepLab V3 [148] +ResNet-50 +- +- +- +- +- +77.8 +DeepLab V3 [148] +ResNet-101 +- +- +- +- +- +78.4 +DeepLab V3plus [148] +MobileNet +- +- +- +- +- +73.8 +DeepLab V3plus [148] +ResNet-50 +- +- +- +- +- +78.8 +DeepLab V3plus [148] +ResNet-101 +- +- +- +- +- +79.6 +Baseline+R.Rotation [148] +ObjectAug +- +- +- +- +- +69.5 +Baseline +R.Scaling [148] +ObjectAug +- +- +- +- +- +70.3 +Baseline + R.Flipping [148] +ObjectAug +- +- +- +- +- +69.6 +Baseline + R.Shifting [148] +ObjectAug +- +- +- +- +- +70.7 +Baseline + All [148] +ObjectAug +- +- +- +- +- +73.8 +Baseline + CutOut (16×16, p = 0.5) [148] +MobileNet +- +- +- +- +- +71.9 +Baseline + CutOut (16×16, p = 1) [148] +MobileNet +- +- +- +- +- +72.3 +Baseline + CutMix (p = 0.5) [148] +MobileNet +- +- +- +- +- +72.7 +Baseline + CutMix (p = 1) [148] +MobileNet +- +- +- +- +- +72.4 +Baseline + ObjectAug [148] +MobileNet +- +- +- +- +- +73.8 +Baseline + CutOut (16×16, p=0.5) + ObjectAug [148] +MobileNet +- +- +- +- +- +73.9 +Baseline + CutMix (p=0.5) + ObjectAug [148] +MobileNet +- +- +- +- +- +74.1 +DeepLabv3+ [14] +EfficientNet-B7 +- +- +- +- +- +84.6 +ExFuse [150] +EfficientNet-B7 +- +- +- +- +- +85.8 +Eff-B7 [156] +EfficientNet-B7 +- +- +- +- +- +85.2 +Eff-L2 [156] +EfficientNet-B7 +- +- +- +- +- +88.7 +Eff-B7 NAS-FPN [37] +EfficientNet-B7 +- +- +- +- +- +83.9 +Eff-B7 NAS-FPN w/ Copy-Paste pre-training [37] +EfficientNet-B7 +- +- +- +- +- +86.6 +TABLE X +RESULTS OF PERFORMANCE (MIOU) ON THE PASCAL VOC 2012 VALIDATION SET + +via style randomization. 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Pseudoseg: Designing pseudo labels +for semantic segmentation. arXiv preprint arXiv:2010.09713, 2020. + diff --git a/1dE0T4oBgHgl3EQf_wJK/content/tmp_files/load_file.txt b/1dE0T4oBgHgl3EQf_wJK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..af73ea20e06cec451a760c3b2ba8e94760cab3a4 --- /dev/null +++ b/1dE0T4oBgHgl3EQf_wJK/content/tmp_files/load_file.txt @@ -0,0 +1,2698 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf,len=2697 +page_content='Advanced Data Augmentation Approaches: A Comprehensive Survey and Future directions Corresponding author(s) 1st Teerath Kumar* ADAPT - Science Foundation Ireland Research Centre and CRT AI, School of Computing, Dublin City University, Dublin, Ireland;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' teerath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='menghwar2@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='dcu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='ie 2nd Muhammad Turab Department of Computer Systems Engineering Mehran University of Engineering and Technology Jamshoro, Pakistan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' turabbajeer202@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='com 3rd Kislay Raj CRT AI, School of Computing, Dublin City University, Dublin, Ireland;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' kislay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='raj2@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='dcu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='ie 4th Alessandra Mileo INSIGHT Centre for Data Analytics and the I-Form Centre for Advanced Manufacturing, School of Computing, Dublin City University, Ireland;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' alessandra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='mileo@dcu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='ie 5th Rob Brennan ADAPT, School of Computer Science, University College Dublin, Ireland;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' rob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='brennan@adaptcentre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='ie 6thMalika Bendechache ADAPT & Lero Research Centres, School of Computer Science, University of Galway, Galway, Ireland;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' malika.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='bendechache@universityofgalway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='ie Abstract—Deep learning (DL) algorithms have shown signif- icant performance in various computer vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' However, having limited labelled data lead to a network overfitting prob- lem, where network performance is bad on unseen data as compared to training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Consequently, it limits performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' To cope with this problem, various techniques have been proposed such as dropout, normalization and ad- vanced data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Among these, data augmentation, which aims to enlarge the dataset size by including sample diversity, has been a hot topic in recent times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' In this article, we focus on advanced data augmentation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' we provide a background of data augmentation, a novel and comprehensive taxonomy of reviewed data augmentation techniques, and the strengths and weaknesses (wherever possible) of each technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' We also provide comprehensive results of the data augmentation effect on three popular computer vision tasks, such as image classification, object detection and semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' For results reproducibility, we compiled available codes of all data augmentation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Finally, we discuss the challenges and difficulties, and possible future direction for the research commu- nity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' We believe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' this survey provides several benefits i) readers will understand the data augmentation working mechanism to fix This publication has emanated from research [conducted with the financial support of supported in part by a grant from] Science Foundation Ireland under Grant number 18/CRT/6223 and is supported by the ADAPT Centre for Digital Content Technology which is funded under the SFI Research Centres Programme (Grant 13/RC/2106/P2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Lero SFI Centre for Software (Grant 13/RC/2094/P2) and is co-funded under the European Regional Development Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' For the purpose of Open Access, the author has applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission overfitting problems ii) results will save the searching time of the researcher for comparison purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' iii) Codes of the mentioned data augmentation techniques are available at 1 iv) Future work will spark interest in research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Index Terms—Big data, Computer vision, Data Augmenta- tion,Deep learning, Image classification, Object detection, Seman- tic segmentation, Survey Data Augmentation I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' INTRODUCTION & MOTIVATION Deep learning models have been very popular and made immense progress in computer vision (CV) tasks such as im- age classification [11], [48], [60], [68], [70], [71], [103], [110], object detection [40], [47], and image segmentation [74], [79], [81], [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' All this advancement has been accelerated by different deep neural network architectures, powerful compu- tation resources, a large amount of accessible data, and mature deep learning libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Among the deep learning models, Convolution Neural Networks (CNNs) have performed well on computer vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' CNNs apply the convolution operation with the input image and kernel to learn different features in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The initial layers of CNN learn the low-level features (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='e edges, lines, etc) while the deep layers learn more structured complex features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The success of CNN has caught the attention to apply it for computer vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Along with CNN, the Vision Transformers (ViT) [28] are also 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='com/kmr2017/Advanced-Data-augmentation-codes arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='02830v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='CV] 7 Jan 2023 getting popular and have been widely used in deep learning for computer vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Although these algorithms are popular and have shown excellent performance in deep learning, they require a lot of data to learn the correct features and avoid overfitting problem [104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Overfitting is when a model is performing well on training data but is not performing on the test (unseen) data, as shown and explained in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' However, data is not always available in large quantities due to various reasons such as privacy issues (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=', medical imaging analysis) or the need for tedious human labeling (object detection, image segmentation) etc [108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Another reason is, it is always tedious, time-consuming and expensive to label data in the case of an availability of unlabeled data [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Even in the availability of huge datasets such as imageNet, data augmentation can still help to reduce overfitting effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It happens because, with a standard training process, the model learns only the important regions (for example head of a dog).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' But it is also necessary for the model to learn other less important features to be more generalized [146].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The CNNs trained on the small set of data often lead to overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Another concern is the adversarial attacks [50], [88], [152], where the noisy perturbation is added to the input image to fool the CNNs and consequently degrade DNNs accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' This modification caused by perturbation is invisible to the human eyes but makes the network fail to identify the correct features in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' To address these problems, data augmentation is mostly applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It is not only useful in computer vision tasks, but also helpful in number of domains such as audio [3], [12], [65], [72], [92], [95], [111], [124] and text domains [6], [33], [84], [109] as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' In this survey, we focus only the computer vision domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Regularization is a technique that generalizes well the model from architectural and data perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' There are several forms of regularization such as Dropout [117], Batch normal- ization [56], transfer learning [107], [137], pre-training [30], data augmentation [154], human-in-the-loop for data aug- mentation [10] and many others [108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Data augmentation is the form of regularization explicitly [69], [110], [154].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Technically, it enlarges the dataset by changing the sample view or flavour [154] to give a diverse view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Other mentioned techniques do not work directly on data like data augmentation as the data is the main cause of any problem for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' If it has an overfitting issue or is biased, it will be propagated to the model as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Carefully performing data augmentation is the key challenge as it is discussed in section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Data aug- mentation is performed on assumption that more information can be extracted from the real dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' But this assumption is not true in real world scienario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Generally, data augmentation solves two key problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' i) the problem of lack of data or limited data, consequently it leads to problem of overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' To solve the overfitting, data augmen- tation makes the model more generalized based on scenario(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' This can be achieved by feeding the various possible scenarios of an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' This indicates more information is extracted indirectly from the original dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' ii) Labeling, the original dataset has a label for each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Augmenting each sample, the label is assigned to the augmented sample as that of the original sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' In some augmentations, the label information is not preserved such as in Mixup data augmentation, labels are also mixed to augment a label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' There are numerous surveys on data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' [98] explores and compares several traditional data augmentations for image classification tasks only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' In another work, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' [130] review available data augmenta- tion approaches for facial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' This work is only limited to face recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Khosla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' [61] discuss warping and oversampling-based data augmentation approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' No taxon- omy, no literature review and no evaluation of the methods are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Shorten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' [108] provide a very detailed work with different aspects of data augmentation, but they did not provide an evaluation of the data augmentation for different CV tasks and they did not include state-of-the-art (SOTA) augmentation methods such as cutmix and grid mask etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Previously the discussed surveys have been two years old and in the last the years, there have been proposed several data augmentation techniques, so it is a dire need for a survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Recently Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' [142] provides a detailed survey with results of several computer vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Very limited results are compiled and SOTA data augmentation methods are not covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Another recently work [141] by Xu, discusses the data augmentation that is model-based and model-free and, proposes novel taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' But this work fails to provide the evaluation of the data augmentation and discusses very limited data augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' There are the number of the data augmentation based on generative adversarial network (GAN) [118], [145], but we do not cover GAN-based data augmentations in this work, as GAN is itslef vast topic and GAN-based techniques are very huge in number Interestingly, none of the mentioned works provides extensive evaluations of SOTA data augmentation and available code compilation based on the proposed taxonomy for result reproducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' To fill these mentioned gaps, our survey makes the following contributions: Presents novel Data augmentation taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Explains SOTA augmentation approaches with visualiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Presents SOTA augmentation evaluation for several tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Compiles the available codes of data augmentations fol- lowing the proposed taxonomy for results reproducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' This survey discusses data augmentation challenges and future directions This survey provides open research questions The above contributions provide the following benefits: A better understanding of data augmentation working mechanism to fix the overfitting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Our comprehensive analysis and comparison between the existing data augmentation techniques will save re- searchers’ time searching this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' facilitate result reproducibility by providing the source code for the different data augmentation techniques in- Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Overfitting problem: On the left side, overfitting is explained in terms of accuracy, after the inflation point (red dotted line), the training accuracy is increasing but validation accuracy is decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' On the right side, alternatively in terms of loss, training loss is decreasing but validation loss is increasing after the red dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The figure is taken from the source 3https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='baeldung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='com/cs/ml- underfitting-overfitting Accuracy Loss Training Validation Validation Training Epochs Epochsvestigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Future work will spark interest in the research commu- nity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' TAXONOMY AND BACKGROUND In this section, we discuss the proposed taxonomy as shown in the figure 2, first data augmentation is classified into two branches, i) basic data augmentations ii) Advanced data augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Then these two are classified further based on operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Background and explanation of each augmentation are discussed below taxonomically: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Basic Data Augmentation Methods This section describes basic data augmentation methods and classifies the augmentation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 1) Image Manipulation: Image manipulation refers to the changes made in an image with respect to its position or color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The positional manipulation is made by adjusting the position of the pixels while color manipulations are made by altering the pixel values of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Image manipulation is further divided into two main categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Each of them is discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Geometric Data Augmentation: Geometric augmentation refers to the changes made with respect to the image geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Geometry refers to position, shifting at certain angle etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' This technique alters the position of pixel values in image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Rotation, Translation, and Shearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Basic geometric augmen- tations are shown in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (i) Rotation : Rotation data augmentation where image is rotated between 0 and 360 degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Degree of rotation is a hyperparameter, it should be chosen wisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Like in case MNIST we can not rotate 180 rotations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' rotation 6 digit by 180 degree, it will be 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' So it won’t make sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It depends on the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (ii) Translation : It is another geometric type data augmen- tation, which shifts the image in upward, downward, right or left direction to give diverse view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The demonstration is shown in the second of the figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (iii) Shearing : Word ‘shear’ means to pervert an image along an axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Shearing is a data augmentation technique that shifts one part of the image to one direction, while the other part is in the reverse direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Technically, it is divided into two categories, x shear and y shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' In x shear, the top part of the image is shifted in one direction and the bottom is shifted in the totally opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' In y shear, the left part of the image is shifted in one direction and the right part is shifted in the reverse direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 2) Non-Geometric Data Augmentations: This category fo- cuses on the visual appearance of the image rather than its ge- ometrical shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Noise injection, flipping, cropping, resizing, and color space manipulation is examples of non-geometric augmentation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Some examples of non-geometric data augmentations are shown in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' A few classical approaches are discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (i) Flipping : it is a kind of data augmentation technique that flips the image either horizontally or vertically, it has shown positive results on the most popular datasets such as cifar10, cifar100 [67] and many more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (ii) Cropping and resizing : Cropping is another data aug- mentation technique that is used as a preprocessing aug- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Either random cropping or central cropping is used as data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' This technique decreases the size of the image then resizing is performed to match the original size of the image, while the labels of the image are not smoothed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (iii) Noise Injection : Injection noise is another technique of data augmentation, that helps neural networks to learn robust features and is quite helpful in defending against adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Nine datasets from the UCI repository have shown impressive results (Reference from [108]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (iv) Color Space: Images having dimensions of H x W x C (where H, w and C represent the height, width and channels, respectively) consist of three channels R, G and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Manipulating each channel values separately in order to control brightness is another way of data aug- mentation, sometimes it is also referred as photometric augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' This augmentation is useful for avoiding the model to be biased toward lightning conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The Simplest way of performing color space augmentation is to isolate any channel and add 2 channels filled with any random value or 0 or 255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Color space is used in photo editing applications i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' to control the brightness or darkness [108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (v) Jitter: It is another data augmentation technique, that randomly changes the brightness, contrast and saturation and hue of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' These four are the hyperparameters and their range (min-max) should be chosen carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' For example, if we increase the brightness of X-Ray images for lung disease detection, it will whiten and mix the lung in X-ray and won’t help disease diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (examples will be shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (vi) Kernel Filters: it is another data augmentation technique that sharpens or blurs the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It starts first, we slide the window of size n x n kernel/matrix of gaussian blur filter or edge filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Gaussian blur filter blurs the image and the edge filter sharpens the edge of the image either horizontally or vertically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 3) Image Erasing Data Augmentations: a) Cutout: It randomly erases the sub region and fills with 0 or 255 in an image during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It shows the impressive performance on very popular datasets [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The demonstration of cutout is shown in figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' b) Random erasing: It [154] randomly erases the sub region in the image like a cutout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' But it also randomly deter- mines to mask out or not and also determines the aspect ratio and size of the masked region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Random erasing demonstration for different tasks is shown in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='c) Hide-and-Seek: The key idea of hide-and-seek data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='augmentation [114] is to divide the image into uniformly ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='squares of random size and randomly remove a random ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Data Augmentations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Basic Data Augmentations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Advanced Data Augmentations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Image Manipulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Image Erasing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Image Mixing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Auto Augment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Neural Style Transfer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Feature Augmentation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Single Image Mixing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Local Augment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='SalfMix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='AutoAugment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Fast AutoAugment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Multi-Image Mixing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Mixup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='CutMix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Geometric ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Manipulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Flipping ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Cropping ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Non-Geometric ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Manipulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Rotation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Translation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Reinforcement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Learning Based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Non-Reinforcement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Learning Based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='FeatMatch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Adversarial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Feature Aug ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Erasing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Cutout ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='GridMask ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='RandAug ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Neural Style ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Style Aug ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='StyPath ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Feature Aug ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Proposed image data augmentation taxonomy Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Overview of the geometric data augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' number of squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It forces neural networks to learn relevant features when important information is hidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' At each epoch, it gives a different view of an image as shown in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' d) GridMask Data Augmentation: GridMask [15] ad- dresses the problem of randomly removing regions that either can completely erase the object or remove context informa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' To do a trade-off between these problems, GridMask is proposed by creating uniform masking and then applying it to images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It is shown in the figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Image Mixing Data Augmentations Image mixing data augmentation has been a hot topic for the last few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Image mixing data augmentation is about mixing image(s) with others or the the same image(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' In this work, we classify the image mixing data augmentation into two categories: Single image mixing Non-single image mixing Single Image Mixing Data Augmentations The single image mixing technique uses only one image and plays around with it from different strategic points of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Recently there has been a lot of work done on single- image augmentation, such as LocalAugment, SelfAugmenta- tion, SalfMix, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The description of each SOTA single image mixing data augmentation has been discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (i) Local Augment: This paper [64] proposes a local aug- ment, that divides image into patches and applies dif- ferent kinds of data augmentation on each with the aim of potentially changing bias properties but generating significant local features, as shown in the figure below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Overview of the non-geometric data augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Random erasing examples for different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Figure source is [154] Input Augmentation Output Rotation Translation ShearingInput Augmentation Output Flipping Cropping & Resize Noise Injection Color Spacing Color Jitterimage classification person re-ID input image Random ErasingAccuracies Method CIFAR10 CIFAR10+ CIFAR100 CIFAR100+ ResNet-18 (Baseline) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='37 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='28 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='32 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='54 ResNet-18 + CutOut 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='69 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='25 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='02 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='58 ResNet-18 + Random Erasing 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='28 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='32 ResNet-18 + CutMix 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='56 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='22 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='58 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='58 ResNet-18 + SaliencyMix 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='41 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='35 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='27 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='71 ResNet-18 + GridMask 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='28 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='54 ResNet-50 (Baseline) 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='86 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='02 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='52 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='42 ResNet-50 + CutOut 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='16 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='14 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='03 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='62 ResNet-50 + CutMix 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='84 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='39 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='35 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='28 ResNet-50 + SaliencyMix 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='19 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='54 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='11 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='43 WideResNet-28-10 (Baseline) [125] 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='03 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='13 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='94 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='20 WideResNet-28-10 + CutOut [27] 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='46 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='92 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='06 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='59 WideResNet-28-10 + Random Erasing 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='2 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='92 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='59 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='27 WideResNet-28-10 + GridMask 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='13 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='24 WideResNet-28-10 + CutMix 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='82 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='13 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='79 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='34 WideResNet-28-10 + PuzzleMix 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='77 WideResNet-28-10 + SaliencyMix 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='96 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='24 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='55 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='44 Note: + sign after dataset name show that traditional data augmentation methods have been used TABLE I BASELINE PERFORMANCE COMPARISON OF VARIOUS AUGMENTATION ON CIFAR10 AND CIFAR100 DATASETS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' An example of Hide-and-Seek augmentation, image is taken from [114] Though this augmentation does not main global structure but provides very diverse features of images, that are essential for neural networks to learn local features in a more generalised way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The visual representation is shown in figure 8 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (ii) Self Augmentation: This paper [106] proposes the self- augmentation, where a random region of an image is cropped and pasted randomly in the image, improves the generalization capability in few-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The process demonstrated in the figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (iii) SalfMix: This paper [20] focuses on whether it is pos- sible to generalize neural networks based on single- image mixed augmentation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' For that purpose, it proposes SalfMix, the first salient part of the image is found to decide which part should be removed and which Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' This figure shows the procedure of GridMask augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' They produce a mask and then multiply it with the input image, the image is taken from [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' portion should be duplicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Most salient regions are cropped and placed into non-salient regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' This process is defined and compared with other techniques in the figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (iv) KeepAugment KeepAugment [41] is introduced to pre- vent distribution shift which degrades the performance of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' KeepAugment’s idea is to increase Training phase W Epoch 1 CNN H Epoch 2 CNN s Training image EpochN CNNCIFAR-10 CIFAR-100 ImageNet Augmentation Accuracy (%) Model Accuracy (%) Model Accuracy (%) Model Cutout [27] 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='04 WRN-28-10 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='59 WRN-28-10 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='1 ResNet-50 Random Erasing [154] 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='92 WRN-28-10 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='27 WRN-28-10 Hide-and-Seek [114] 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='53 ResNet-110 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='13 ResNet-110 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='20 ResNet-50 GridMask [15] 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='24 WRN-28-10 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='9 ResNet-50 LocalAugment [64] 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='92 WRN-22-10 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='87 ResNet-50 SalfMix [20] 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='62 PreActResNet-101 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='11 PreActResNet-101 KeepAugment [41] 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8 ResNet-28-10 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='3 ResNet-101 Cut-Thumbnail [140] 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8 ResNet-56 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='94 WRN-28-10 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='21 ResNet-50 MixUp [147] 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='3 WRN-28-10 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='5 WRN-28-10 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='9 ResNet-50 CutMix [146] 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='10 WRN-28-10 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='40 WRN-28-10 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6 ResNet-50 SaliencyMix [125] 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='24 WRN-28-10 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='44 WRN-28-10 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='74 ResNet-50 PuzzleMix [63] 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='05 WRN-28-10 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='51 ResNet-50 FMix [45] 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='64 Pyramid 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='95 Dense 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='70 ResNet-101 MixMo [101] 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='38 WRN-28-10 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='40 WRN-28-10 StyleMix [52] 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='44 PyramidNet-200 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='83 PyramidNet-200 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='29 PyramidNet-200 RandomMix [85] 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='02 WRN-28-10 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='84 WRN-28-10 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='88 WRN-28-10 MixMatch [9] 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='05 WRN-28-10 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='12 WRN-28-10 ReMixMatch [8] 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='71 WRN-28-2 FixMatch [115] 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='69 WRN-28-2 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='04 WRN-28-2 AugMix [49] 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6 ResNet-50 Improved Mixed-Example [120] 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='02 ResNet-18 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='3 ResNet-18 RICAP [122] 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='18 WRN-28-10 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='56 ResNet-28-10 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='62 WRN-50-2 ResizeMix [100] 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='60 WRN-28-10 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='31 WRN-28-10 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='00 ResNet-50 AutoAugment [23] 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='40 WRN-28-10 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='90 WRN-28-10 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='50 AmoebaNet-C Fast AutoAugment [82] 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='00 SS(26 2×96d) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='10 SS(26 2×96d) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='60 ResNet-200 Faster AutoAugment [46] 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='00 SS(26 2 × 112d) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='40 SS(26 2×96d) 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='90 ResNet-50 Local Patch AutoAugment [83] 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='10 SS(26 2 × 112d) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='90 SS(26 2×96d) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='00 ResNet-200 RandAugment [24] 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='50 PyramidNet 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='30 WRN-28-10 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='00 EfficientNet-B7 TABLE II PERFORMANCE COMPARISON OF THE VARIOUS IMAGE ERASING AND IMAGE MIXING AUGMENTATIONS FOR IMAGE CLASSIFICATION PROBLEMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' WRN STANDS FOR WIDERESNET AND SS FOR SHAKE-SHAKE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' An example of Global and Local Rotation Image, example is taken from [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Comparison of LocalAugment with CutOut, MixUp etc, example is taken from [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' fidelity by preserving the salient features of the image and augmenting the non-salient region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Preserved features further allow for increased diversity without shifting the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Keep augment is shown in the figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (v) You Only Cut Once You Only Cut Once (YOCO) [44] is introduced with the aim of recognizing objects from partial information and improving the diversity of augmentation that encourage neural networks to perform Input Image GlobalRotation 1 LocallySegmented LocalRotation Input ImageInput Image Cutout Random Erasing Mixup CutMix Local AugmentFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' An example of self augmentation, image is taken from [106] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Conceptual comparison between SalfMix method and other single image-based data augmentation methods, example is taken from [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' This image shows the example of KeepAugment with other augmentations, courtesy [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' YOCO makes two pieces of image and augmen- tation is applied one each piece, then each piece is concatenated for an image and YOCO shows impressive performance and compared with SOTA augmentations, sometimes it outperforms them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It is easy to implement, has no parameters, and is easy to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The YOCO augmentation process is shown in the figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' An example of YOCO augmentation, image is taken from [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (vi) Cut-Thumbnail : Cut-Thumbnail [140] is a novel data augmentation, that resizes the image to a certain small size and then randomly replaces the random region of the image with the resized image, aiming to alleviate the shape bias of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The advantage of Cut- thumbnail is, that it not only preserves the original image but also keeps it global in the small resized image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' On Im- ageNet, it shows impressive performance using resnet50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Overall, the cut-thumbnail process and its comparison are shown in figure 15 and figure 14, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Comparison between existing data augmentation methods with Cut- Thumbnail, example is from [140].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' This image shows an example of reduced images that is called thumbnails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' After reducing the image to a certain size 112×112 or 56×56, The dog is still recognizable even though lots of local details are lost, courtesy [140].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Non-Single Image Mixing Data Augmentations Non-Single image mixing data augmentation uses more than one image and applies different mixing strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Recently, many researchers explored a lot of non-single image mixing strategies and still, it is a very attentive topic for many researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Recently work has included Mixup, CutMix, Scale Crop Flip Cutout Single Training Image Saliency Map Self-mixed Image (al) Red fox (a2) Cutout (a3) RandAugment (a4) Saliency map (a5) Keep+Cutout (a6) Keep+RandAugment>S Aug Concat Cut Aug(a) Original Sample (b) Cutout (c) Mixup (d) CutMix (e) Cut-Thumbnail224SaliencyMix, and many more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Each of the relevant non- single image mixing data augmentation techniques is discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (i) Mixup: It blends any random two images based on the blending factor (alpha) and the corresponding labels of these images are also mixed in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Mixup data augmentation [147] sustainable improved the perfor- mance not only in terms of accuracy but also in terms of robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Experiments on ImageNet-2012, CIFAR-10, CIFAR-100, Google commands and UCI datasets showed impressive results on SOTA methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' As it is compared and shown in the figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (ii) CutMix : It [146] tackles the issues of information loss and region dropout issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It is inspired by cutout [27], where any random region is filled with 0 or 255, while in cutmix instead of filling the random region with 0 or 255, the region is filled with a patch from another image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Cor- respondingly their labels are also mixed proportionally to the number of pixels mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (as shown in figure 16) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Overview of the Mixup, Cutout, and CutMix, example is from [146].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (iii) SaliencyMix : It [125] basically addresses the problem of cutmix and argues that filling a random region of the image with a patch from another won’t guarantee that patch has rich information and thereby mixing labels of unguaranteed patches leads the model to learn unneces- sary information about the patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' To deal with that issue, saliencyMix first selects the salient part of the image and pastes it to a random region or salient or non-salient of another image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (as shown in figure 17 and figure 18) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' An example of SaliencyMix augmentation, image is taken from [125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (iv) Puzzle Mix : This article [63] proposes a puzzle mix data augmentation technique that focuses on using explicitly salient information and basic statistics of image wisely with the aim of breaking misled supervision of neural networks over existing data augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Furthermore, the demonstration is shown and compared with relevant methods in the figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (v) SnapMix: The article [53] proposes the Semantically Proportional Mixing (SnapMix) that utilises class activa- Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' This image shows the proposed SaliencyMix data augmentation procedure, courtesy [125] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' A visual comparison of the mixup methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Puzzle Mix ensures to contain sufficient target class information while preserving the local statistics of each in, example is from [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' tion map (CAM) to reduce the label noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' SnapMix creates the target label considering the actual salient pixel taking part in the augmented image, which ensures semantic correspondence between the augmented image and mixed labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The overall process is demonstrated and compared with closely matching augmentations in the figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' A visual Comparison of Mixup, CutMix, and SnapMix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The figure gives an example where SnapMix’s generated label is visually more consistent with the mixed image’s semantic structure comparing to CutMix and Mixup, courtesy [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (vi) FMix: This article proposes the FMix [45], a kind of mixed sample data augmentation (MSDA), utilises the random binary masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' These random binary masks are acquired by applying a threshold to low-frequency images Target Image Source Image Augmented Image 20% Mixed label for randomly mixed images Dog - 80% & Cat 20% ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' D0g - 80% & Cat 20% ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Selecting the Peak Selecting the Mixing the Source Source Image Saliency Map of Salient Region of Source Patch Based Target Image Patch with the Augmented Image the Source Image on the Peak Salient the Saliency Map Target Image RegionInput1 Input Mixup Puzzle Mix (z only) Input2 CutMix Puzzle Mix (full)MixUp SnapMix 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='4x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6x Ya: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='4, Yb: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6 Asymmetrical Mixture Ya Yb Ya CutMix Ya: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6, Yb: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='4 Ya Ya: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='28, Yb: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='72 Semantic-relatedness Proportionthat are obtained from Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Once the mask is obtained, one colour region is applied on input one and another colour region is applied on another input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The overall process is shown in figure 21: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Example masks and mixed images from CIFAR-10 for FMix, example is from [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (vii) MixMo : This paper [101] focuses on the learning of multi-input multi-output via subnetwork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Main motivation of the paper is to replace direct hidden summing oper- ations with more solid mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' For that purpose, it proposes MixMo, which embeds M inputs into shared space, mixes them and passes them to a further layer for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Moreover, the overall process is demon- strated in figure 22: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' This image shows the overview of MixMo augmentation, image is taken from [101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (viii) StyleMix : This paper [52] targets previous approaches problems, they don’t differentiate between content and style features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' To remedy this, this problem proposes two approaches styleMix and StyleCutMix, this is the first work that separately deals with content and style features of images very carefully and it showed impressive per- formance on a popular benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The overall process is defined and compared with SOTA approaches in the figure 23: (ix) RANDOMMIX : This paper [85] improves generaliza- tion capability by proposing randomMix, which randomly selects mix augmentation from a set of augmentations and applies it to images, enabling the model to look at diverse Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' A Visual comparison of StyleMix [52] and StyleCutMix with Mixup [147] and CutMix [146], example is from [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' This method showed impressive results over SOTA image mixing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The overall demonstration is shown in the figure 24: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' An illustrative example of RandomMix, image is taken from [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (x) MixMatch : Data augmentation technique is very useful in semi-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' MixMatch [9] augments single image K time and passes all K number of images to a classifier, averages their prediction and finally, their predictions are sharpened by adjusting their distribution temperature term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (as shown in the figure 25) (xi) REMIXMATCH : This work [8] is an extension of mix match and makes prior work efficient by introduc- ing distribution alignment and augmentation anchoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Mask Image 1 Image 2 FMixNetwork Λy Co do MixMo Mixing d Sum Mixing (MIMO: 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='06 %) Patch Mixing (Our Cut-MixMo: 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='40 %) orInput 1 Input 2 Mixup StyleMix CutMix StyleCutMix Method Content Parrot 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='5 Parrot 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='4 Parrot 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='2 Parrot 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='2 label Panda 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='5 Panda 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6 Panda 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8 Panda 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8 Style Parrot 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8 X Parrot 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6 X label Panda 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='2 Panda 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='4Input batch Random Pairs(Input batch, randperm(Input batch)) sample pairs random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='choices(Candidates,Weights).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Candidates=[Miaup,CutMia,ResizeMir,Fmia], Random Weights = [1,1,1, 1] mixing method Mirup OR CutMia OR ResizeMia OR Fmia Random wBeta(a,a w U(0, 1) 入U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8) wBeta(Q,a mixing ratio OR OR OR Mixup CutMix ResizeMix Fmix Cat (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='52) Cat (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='21) Cat (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='15) Cat (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='39) Dog (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='48) Dog (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='79) Dog (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='85) Dog (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='61)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Diagram of the label guessing process used in MixMatch, courtesy [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Distribution alignment tasks are to make the marginal distribution of predictions on unlabeled data close to the marginal distribution of ground truth and encourage the marginal distribution of predictions on unlabeled data to be close to the marginal distribution of ground truth labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Augmentation anchoring feeds multiple strongly augmented versions of an input into the model and encourages each output to be close to the prediction for a weakly-augmented version of the same input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It is shown in figure 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Anchoring augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It makes predictions on strong augmen- tations of the same image (blue) using the forecast for a weakly enhanced image (green, centre), courtesy [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (xii) FixMatch : Fixmatch [115] also alleviates the perfor- mance of semi-supervised learning (SSL), the model is trained on limited labeled data then the trained model is used to assign the label to unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fixmatch first assigned pseudo labels to unlabeled images having a probability higher than a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The model is forced to make predictions on a strong augmented version of the unlabeled image to match its prediction with the pseudo label using cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (Overall process is shown in the figure 27) (xiii) AugMix : Augmix [49] is a simple and effective data augmentation that reduces the gap between the distri- bution of training and test (unseen) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' M operations are performed with a corresponding random magnitude of augmentation and at the end, all those images are merged to produce a new image that widely explores the semantically equivalent input space around an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' As shown in the figure below, three operations are performed separately in three branches and further operations are also performed for diversity purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Finally, all images Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' This image shows the procedure of FixMatch, image is taken from [115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' are mixed to generate a new image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It is very useful for robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It is shown in figure 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' An overall procedure of AugMix augmentation [49], example is from [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (xiv) Simple Copy-Paste is a Strong Data Augmentation Method for Instance Segmentation : This method [37] simply copies and pastes the instances of one image to another image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It shows promising results and is very easy to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' As shown in the figure below, two images’ instances are pasted to each other on different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It is visually shown in figure 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Image augmentation performed by simple Copy-Paste [37] method, courtesy [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (xv) Improved Mixed-Example Data Augmentation: These days state-of-the-art non-label preserving data augmenta- tion techniques have shown promising results using linear combinations of the two examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' This paper [120] ex- plores research questions: i) Why do these methods work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' ii) By proposing new augmentations, is this linearity important?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It is shown in figure 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Classify .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' K augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Average Sharpen Unlabeled ClassifyWeakly- augmented Prediction Pseudo-label Unlabeled Model example Strongly- augmented Prediction Modelshear_y Xorig Wj=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='12 Xaugmix trans Xaug W2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='2 1-m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8 rotate post teriz m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='2 equalize posterizecopy-pasteFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' A visual comparison of linear methods and generalized augmentation performed by Improved Mixed-Example, image is taken from [120].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (xvi) RICAP : Random image cropping and patching (RI- CAP) [122] is a new data augmentation technique that cuts and mixes four images rather than two images, and the labels of the images are also mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It shows impressive performance on popular datasets i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' CIFAR10 , CIFAR100, and imageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' For more detail, RICAP is shown in the figure 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' A conceptual explanation of the RICAP data augmentation, the example is from [122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (xvii) Rethinking Data Augmentation for Image Super- resolution: A Comprehensive Analysis and a New Strategy : This paper [143] explores and analyses ex- isting data augmentation techniques for super-resolution and proposes another data augmentation technique for super-solution, named cutblur that cuts high-resolution image patches and pastes to corresponding low-resolution images and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Cutblur shows impressive perfor- mance on super-resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Furthermore, the process is illustrated in the figure 32 and 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' An Schematic illustration of CutBlur operation, image is taken from [143].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' A visual comparison between High resolution, low resolution and CutBlur, courtesy [143].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (xviii) ResizeMix: Mixing Data with Preserved Object Infor- mation and True Labels : The ResizeMix [100] method directly cuts and pastes the source data in 4 different ways to target the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 4 different ways including salient part, non-part, random part or resize source image to patch, as shown in the figure 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It addresses two questions: How to obtain a patch from the source image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' where to paste the patch from the source image in the target image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Furthermore, it was found that saliency information is not important to promote mixing data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' ResizeMix is shown in the figure 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (xix) ClassMix: Segmentation-Based Data Augmentation for Semi-Supervised Learning : This research work [93] proposed novel data augmentation for semi-supervised semantic segmentation with inspiration, traditional data augmentation is not effective for semantic segmentation as they are for image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Proposed data aug- mentation named ClassMix, which augments the training sample by mixing unlabeled samples, by exploiting net- work prediction while considering object boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The proposed approach showed signification performance on two common datasets for semi-supervised semantic seg- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The overall process is shown in the figure 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (xx) Context Decoupling Augmentation for Weakly Su- pervised Semantic Segmentation : This article [119] Input Linear Methods GeneralizedCut-and-paste Cut-and-paste LR : HR : LR HR HR LR LR HR (input) HR LR (input)HRFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' A visual representation of different cropping manners from the source image and different pasting manners to the target image, image is taken from [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' In a visual representation classMix augmentation, two images are sampled then based on the predictions of each image a binary mask is created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The mask is then used to mix the images and their predictions, the image is taken from [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' addresses the problem of traditional data techniques for WSSS, increasing the same contextual data semantic samples does not add much value in object differentia- tion, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' image classification, “cat” recognition is due to the cat itself and also its surrounding context, that discourages model to focus only on the cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' To break this, this article proposed a novel data augmentation named Context Decoupling Augmentation, to make it diverse where the specific object appears and guide the network to break the dependencies between object and contextual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' In this, the way it also provides augmenta- tion and the network focus to object instance rather than object instance and contextual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' A comparison of traditional data augmentation and Context Decoupling Augmentation is shown below in the figure 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (xxi) ObjectAug: Object-level Data Augmentation for Se- Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' A visual representation of the difference between the conventional augmentation approach and context decoupling augmentation (CDA), image is taken from [119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' mantic Image Segmentation : This article [148] ad- dresses the problem of mixing image-level data augmen- tation strategies, which failed to operate for segmentation since at object and background are coupled as bound- aries of objects are not augmented due to their fixed semantic bond with the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' To mitigate this problem, this article proposes a novel approach named ObjectAug, object-level augmentation for semantic seg- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' First, it separates object(s) and backgrounds from an image with the help of semantic labels then each object is augmented using popular data augmentation techniques such as flipping and rotating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Pixel changes due to these data augmentations are restored using image inpainting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' In the end, the object(s) and background are coupled to create an augmented image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Experimental re- sults suggest that ObjectAug has shown effective perfor- mance improvement for segmentation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Furthermore, ObjectAug is shown in the figure 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Advance Data Augmentation Methods AutoAugment: The goal of this technique is to find the data augmentation policies from training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It solves the problem of finding the best augmentation policy as a discrete search problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It consists of a search algorithm and a search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It is divided into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Reinforcement learning data augmentation Non-Reinforcement learning data augmentation 1) Reinforcement Learning data augmentations: Rein- forcement learning data augmentaion technique generalize and improve the performance of deep networks in an environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (i) AutoAugment : This work [23] automatically finds the best data augmentation rather than manual data augmen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' To address this limitation, this article proposes autoaugment, where search space is designed and has policies consisting of many sub-policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Each subpolicy has two parameters one is image processing function and Source patch P Source image Is Salient region Target image It iNon-salient region How to obtain?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Where to paste?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Random region Resizefe Conventional Augmentation Rotation Color jittering Raw image CDAFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' ObjectAug can perform various augmentation methods for each object to boost the performance of semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The left husky is scaled and shifted, while the right one is flipped and shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Thus, the boundaries between objects are extensively augmented to boost their performance, the example is from [148].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' the second one is the probability with magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' These subpolicies are found using reinforcement learning as a search algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The overall process is shown in the figure 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' A visual overview of the sub-policies from ImageNet using AutoAugment, example is from [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (ii) Fast AutoAugment : Fast Autoaugment [82] addresses the problem of autoaugment, it takes a lot of time to find the optimal data augmentation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' To end this, fast auto augment finds more optimal data augmentations using an efficient search strategy based on density match- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It reduces the higher order of training time compared to auto augment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The overall procedure is shown in figure 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (iii) Faster AutoAugment: This article proposes a faster autoaugment [46] policy intending to find effective data augmentation policies very efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Faster autoaug- ment is based on a differentiable augmentation searching policy and additionally, it not only estimates gradients for many transformation operations having discrete pa- rameters but also provides a mechanism for choosing operations efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Moreover, it introduces a training Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' An overall procedure of augmentation search by Fast AutoAugment algorithm, courtesy [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' objective function with aim of minimising the distance between original and augmented distribution, that is also differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Parameters of augmentations are updated during backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The Overall process is defined in figure 40: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' An Overview of the Faster AutoAugment augmentation, image is taken from [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (iv) Reinforcement Learning with Augmented Data: This paper proposes Reinforcement Learning with Augmented Data (RAD) [76], easily pluggable and enhances the performance of RL algorithms by targeting two issues i) learning data efficiency ii) generalisation capability for new environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Furthermore, it shows traditional data augmentation techniques enable RL algorithms to outperform complex SOTA tasks for pixel-based control and state-based control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Overall process is defined in (a)Original image (b) Original label (c) Augmented image (d) Augmented labelOriginal Sub-policy 1 Sub-policy 2 Sub-policy3 Sub-policy 4 Sub-policy5 Batch 1 Batch2 Batch3 Equalize, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='4, 4 Solarize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6,3 Posterize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='5 Rotate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='3 Equalize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8 Rotate, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8,8 Equalize,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6,7 Equalize, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='0, 2 Solarize,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6,8 Posterize,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='4,6(K) D(K) sample B select N D(K) M train evaluate (y)Q M(0) train split +++ (1) sample B select N Augment Policy D() M train evaluate apply D(1) train M(0) T(Dtrain) trainOriginal or Classified Augmented?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' + correctly?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Critic Policy Update policy by backpropagationfigure 41: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' An overview of different augmentation investigated in RAD, example is from [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (v) LOCAL PATCH AUTOAUGMENT WITH MULTI- AGENT COLLABORATION: This is the first paper [83] that finds data augmentation policy for patch level using reinforcement learning, named multi-agent rein- forcement learning (MARL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' MARL starts by dividing images into patches and jointly finds optimal data aug- mentation policy for each patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It shows competitive results on SOTA benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' MARL is compared and differentiated with other augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Overall process is defined in figure 42: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' An Illustration of different automated augmentation policies, courtesy [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (vi) Learning Data Augmentation Strategies for Object Detection: This work [155] proposes to use autoaugment that learns the best policies for object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It finds the best value and then compares it with the value of architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It addresses two key issues of augmentation for object detection, a) Classification learned policies can not directly be ap- plied for detection tasks, and it adds more complexity to deal with bounding boxes in a case if geometric augmentations are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' b) Most research thinks it adds much less value compared to designing new network architecture so gets less attention but augmentation for object detection should be selected carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Some sub-policies for this data augmentation are shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Different data augmentation sub-policies explored, image is taken from, [155].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The sub-policies details are given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Sub-policy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (Color, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='2, 8), (Rotate, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8, 10) Sub-policy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (BBox Only ShearY, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8, 5) Sub-policy 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (SolarizeAdd, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6, 8), (Brightness, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8, 10) Sub-policy 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (ShearY, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6, 10), (BBox Only Equalize, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6,8) Sub-policy 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (Equalize, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6, 10), (TranslateX, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='2, 2) (vii) Scale-aware Automatic Augmentation for Object De- tection: This work [18] proposes a new data augmenta- tion for object detection named scale aware autoAug, first, it defines a search space where image level and box level data augmentation are prepared for scale invariance, sec- ondly, this work also proposes a new search metric named Pareto scale balance for search augmentation effectively and efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Some examples of data augmentation are shown in figure 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (viii) ADA: Adversarial Data Augmentation for Object Detection: Data augmentation for object detection has improved performance but it is difficult to understand whether these augmentations are optimal or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' This Input Crop Translate Window Grayscale Cutout Flip Rotate Cutout-color Random conv Color-jitter Augmentations applied consistentlyacross stacked framesOriginal Images Image-Wise AutoAugment Patch-Wise AutoAugmentBatch 1 Batch 2 Batch 3 Batch 4 Sub-policy 1 Sub-policy 2 Sub-policy 3 Sub-policy 4 Sub-policy 5Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Example of scale-aware search space which includes image level and box-level augmentation, the example is from, [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' article [7] provides a systematic way to find optimal adversarial perturbation of data augmentation from an object detection perspective, that is based on game- theoretic interpretation aka Nash equilibrium of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Nash equilibrium provides the optimal bounding box pre- dictor and optimal design for data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Optimal adversarial perturbation refers to the worst perturbation of ground truth, that forces the box predictor to learn from the most difficult distribution of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' An example is shown in figure 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Annotation distribution types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Adversarial augmentation chooses bounding boxes that are as distinct from the truth as possible while yet containing crucial object characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The example is from, [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (ix) Deep CNN Ensemble with Data Augmentation for Ob- ject Detection: This article [42] proposes a new variant of the R-CNN model with two core modifications in training and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' First, it uses several different CNN mod- els as ensembler in R-CNN, secondly, it smartly augments PASCAL VOC training examples with Microsoft COCO data by selecting a subset from Microsoft COCO datasets that are consistent with PASCAL VOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Consequently, the dataset size is enlarged and improves the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The schematic diagram is shown in the figure 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (x) Robust and Accurate Object Detection via Adversarial Learning: This [16] first shows classifier performance gain from different data augmentations when fine-tuned to object detection tasks disappears and performance in terms of accuracy and robustness is not improving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The article provides a unique way of exploring adversarial samples that helps to improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' To do so, it augments the example during the fine-tuning stage for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The proposed schematic diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Example is from, [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' object detectors by exploring adversarial samples, which is considered model-dependent data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' First, it picks the stronger adversarial sample from detector classification and localization layers and these change with the detector to ensure augmentation policy remains consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It showed significant performance gain in terms of accuracy and robustness on different object detection tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (xi) Perspective Transformation Data Augmentation for Object Detection: This article [129] proposes a new data augmentation for objection detection named perspec- tive transformation that generates new images captured at different angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Thus, it mimics images as if they are taken at a certain angle where the camera can not capture those images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' This method showed effectiveness on several object detection datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' An example of the proposed data augmentation is shown in the figure below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (xii) Deep Adversarial Data Augmentation for Extremely Low Data Regimes: This article [149] addresses the issue of extremely low data regimes: labeled data is at a very low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' To deal with that problem, it proposes a deep adversarial data augmentation (DADA), where data augmentation is formulated as a problem of training class conditional and supervised GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Furthermore, it also introduces new discriminator loss with aim of fitting data augmentation were real and augmented samples are forced to participate equally and be consistent in finding decision boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 2) Non-Reinforcement Learning data augmentations: dummy text here (i) RandAugment: Previous optimal augmentation finding uses reinforcement or some complex learning strategy that takes a lot of time to find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' RandAugment augmen- tation [24] removes obstacles of a separate searching phase, which makes training more complex and conse- Image-level Aug Sample images by Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Box-level Aug Box-levelAugpolicy contains Mag is for zooming ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Zoom-in/out-(Prob,Mag) Color/Geometric-(Prob,Mag,Area) ColorandGeometricoperations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Aug types Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Area ratio Brightness P1 M1 Color P2 M2 Zoom-out Largeobject Contrast P3 M3 W S0n≥0 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' r(Sbox) <1 Cutout P4 M4 Color r(Sbox) Equalize Ps Ms Sharpness Ps M6 Solarize P, M7 H SolarizeAdd P: Mg Hflip Pg Mg Rotate P10o M1o Original Shearx P11 M11 Geometric r(Sbox) Pori= 1 - Pin - Pout Middleobject Sheary P12 M12 W e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' r(sbox) > 1 TranslateX P13 M13 Translatey P14 M14 r(Sbox) is the ratio of aug area and box size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' H It varies fordifferent scale of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Zoom-in Smallobject "o e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' r(sbox) > 1 Scale-awareBox-levelAug Bounding box AugmentedareaSingle Ground Truth RandomAugmentation AdversarialAugmentationPrediction Average VGG-16 GoogleNet VGG-16 GoogleNet VGG-16 GoogleNet PASCAL2012 PASCAL2007 PASCAL2012 coco Filterec PASCAL2007 PASCAL2012 COCO2014Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Overview of Robust and Accurate Object detection via adversarial learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' In the top image, it improves object detector accuracy on clean im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' In middle, improves the detector’s robustness against natural corruption, and at the bottom, it improves the robustness against cross-dataset domain shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The image is taken from, [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' quently adds computational cost overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' To break this, randaugment random applies N data augmentations with M magnitude of all augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Some visualisation is demonstrated in the figure 48: 3) Neural Style Transfer:: It is another category of data augmentation, which can transfer the artist style of one image to another without changing semantics at a high level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It brings more variety to the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The main objective of this neural style transfer is to generate a third image from two images, where one image provides texture content and another provides high-level semantic content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (i) STaDA: Style Transfer as Data Augmentation : This work [153] thoroughly evaluated different SOTA neural style transfer algorithms as data augmentation for image classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It shows significant performance gain on Caltech 101 and Caltech 256 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Furthermore, it also combines neural style transfer algorithms with conventional data augmentation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' A sample of this augmentation is shown in figure 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Example images augmented by RandAugment, image is taken from [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Overview of the original image and two stylized images by STaDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Image is taken from, [153].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (ii) Neural Style Transfer as Data Augmentation for Improving COVID-19 Diagnosis Classification : This work [51] shows the effectiveness of a cycle generative adversarial network (GAN), which is mostly used for neural style transfer, augments COVID-19 negative x-ray image to convert into positive COVID image to balance the dataset and also to increase the diversity of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It shows that augmenting the images with Cycle GAN can improve performance over several different CNN architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' A sample of this augmentation is shown in figure 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (iii) Style Augmentation: Data Augmentation via Style Randomization: This work [59] proposed a novel data augmentation named style augmentation (SA) based on style neural transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' SA randomizes the color, contrast, and texture while maintaining the shape and semantic content during the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' This is done by picking an arbitrary style transfer network for randomizing the style and by getting the target style from multivariate normal distribution embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It improves performance in three different tasks: classification, regression, and Vanilla Det-AdvProp (ours) potted plant: 47% pottedplant:37% spoon:32% person:98% person: 93% bowl:35% ovenz 48% oven: 55% 47% oven: 70% person: 49% knife: 32% bowl: 41% 90wl:50% bowl: spoon:49% bowl: 67% COCO (+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='3~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='1 mAP) Accurateon Clean Images person: 76% person: 79% person:40% oven:39% oven:37% bowl:34% COCO-C(+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8mAP) Robust to Natural Corruption potted plant: 44% %06 potted plant: 51% cat:93% PASCALVOC (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='2~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='3mAP) RobusttoDomain ShiftMagnitude:9 Original Shearx AutoContrast Magnitude:17 Original Shearx AutoContrast Magnitude:28 Original Shearx AutoContrast(a) Original (b) Snow (c) YourNameFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Overview of generating synthetic covid images from the healthy category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' As the no of epochs grows the quality of the synthetic images improves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Example is from [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The style augmentation sample is shown in figure 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Overview of Style augmentation applied to an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The shape is preserved but the style, including color, texture, and contrast is randomized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Image is from [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (iv) StyPath: Style-Transfer Data Augmentation for Ro- bust Histology Image Classification: This paper [22] proposes a novel pipeline for Antibody Mediated Rejec- tion (AMR) classification in kidneys based on StyPath data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' StyPath is data augmentation that transfers style intending to reduce bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The proposed augmentation is much faster than SOTA augmentations for AMR classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Some samples are shown in figure 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (v) A Neural Algorithm of Artistic Style : This work [36] introduces an artificial system (AS) based on Deep neural network that generates artistic images of high perceptual quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' AS creates neural embedding and then AS uses the embedding to separate the style and content of the image and then recombines the content and style of target images to generate the artistic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The sample is shown in figure 53 4) Feature space data augmentations: Feature data aug- mentation is another category of data augmentation, where first images are first transformed into embedding or representation then data augmentation is performed on the embedding of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Recently a few works have been done in this area, we selectively highlight the work in a precise way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Comparison of content and random initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Authors observe that output images initialized as noise appeared distorded and discolored and failed to retain the content fidelaty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Image is from [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Overview of the styled image by neural algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Image is from [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (i) Dataset Augmentation in Feature Space : This work [26] first used encoder-decoder to learn representation, then on representation apply different transformations such as adding noise, interpolating, or extrapolating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The proposed method has shown performance improvement on both static and sequential data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Overview of interpolation and extrapolation between handwritten characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Original characters are shown in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Image is taken from [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (ii) Feature Space Augmentation for Long-Tailed Data : This paper [21] proposed the novel data augmentation in feature space to address the long-tailed issue and uplift the under-represented class samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The proposed ap- proach first separates class-specific features into generic and specific features with the help of class activation maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Under-represented class samples are generated by injecting class-specific features of under-represented classes with class-generic features from other confusing Epochs 5 10 15 20 25 3010 iter 50 iter 100 iter 100 iter X cont init noise_init X X X sty ino inoQ a a 8 8 a @ 8 8 Q d (a) Interpolation (b) Extrapolationclasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' This enables diverse data and also deals with the problem of under-represented class samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It has shown SOTA performance on different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' As it is demonstrated in figure 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Left: limited but well-spread data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Right: Without sufficient data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Image is taken from [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (iii) Adversarial Feature Augmentation for Unsupervised Domain Adaptation: Generative Adversarial Networks (GANs) showed promising results in unsupervised do- main adaptation to learn target domain features indis- tinguishable from the source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' This work [127] extends GAN to force features extractor to be domain- invariant ii) To train it via data augmentation in feature space, named feature augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' This work explores data augmentation at the feature level with GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (iv) Understanding data augmentation for classification: when to warp?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' : This paper [138] investigates the data augmentation advantages on image space and feature space during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It proposed two approaches i) data warping which generates extra samples in image space using data augmentations and ii) synthetic over- sampling, which generates samples in feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It also suggests that it is possible to apply general data augmentation techniques in feature space if reasonable data augmentations for data are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' (v) FeatMatch: Feature-Based Augmentation for Semi- Supervised Learning : This work [73] presents a novel approach of data augmentation in features space for SSL inspired by an image-based SSL method that uses a com- bination of augmentations of the images and consistency regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Image-based SSL methods are restricted to only conventional data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' To break this end, the feature-based SSL method produced diverse features from complex data augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' One key point is, these advanced data augmentations exploit the information from both intra-class and inter-class representations ex- tracted via clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The proposed method only showed significant performance gain on min-Imagenet such as an absolute 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='44% gain on miniImageNet, but also showed robustness on samples that are out-of-distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' More- over, the difference between image-level and feature-level augmentation and consistency is shown in figure 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' An overview of featMatch augmentation applied on images and features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Image is taken from [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' RESULTS In this section, we provide the detailed result for various CV tasks such as image classification, object detection, and semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The main purpose is to show the effect of the data augmentation in CV different tasks and to do so, we compile results from various SOTA data augmentation works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Image Classification In this section, we present the result of several SOTA data augmentation methods for supervised learning and semi- supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Both are discussed below: 1) supervised learning results: Supervised learning is, we have data on a large quantity that is wholly labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' We train NN on that data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' In this section, we compare and compile the results from several SOTA data augmentation methods and put them in two different tables as shown in table II-A3d and table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' In table II-A3d results, + sign shows traditional data augmentations such as flipping, rotating and cropping, have been used along with SOTA augmentation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The used datasets are CIFAR10, CIFAR100 and ImageNet, and the used networks are wideresnet flavours, pyramid network flavours and several popular resnet flavours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' All the classification results are reported in accuracy, the higher is the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' As it is noticed from table II-A3d and table II that each data augmentation has significantly improved the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 2) Semi-supervised learning: Semi-supervised learning (SSL) is when we have limited labeled data but unlabeled data is available on large scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Labeling the unlabeled data is tedious, time-consuming and cost [71], [139].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' To avoid these issues, SSL is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' There are several techniques of SSL, but recently data augmentation is employed with the limited labeled data to increase the diversity of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Data augmentation with SSL has increased the performance on different datasets and NN architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The used dataset are CIFAR10, CIFAR100, SVHn and Mini-ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Several Learned Boundary Learned Boundary True True Boundary Boundary Re-weight Samplefc l softmax x Encoder layer fx p(ylf) Image-based Consistency Loss Augmentation fc layer softmax x Encoder fx p(ylfx) (a) Image-Based Augmentation and Consistency fc layer softmax gx Prototypes per class ★ p(ylgx) Feature-based Consistency Loss Augmentation fc l softmax Encoder layer x fx p(ylfx) (b) Feature-Based Augmentation and ConsistencySSL techniques are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' We compile the results from many SOTA SSL methods with data augmentation and present in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The effect of the data augmentation has also been shown with the different number of samples in SSL as shown in table III, table IV and table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Object detection In this section, we discuss the effectiveness of various image data augmentation techniques on the frequently used COCO2017, PASCAL VOC, VOC 2007, and VOC 2012 datasets, which are commonly used for object detection tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' We compile results from various SOTA data augmentation methods and put them in three different tables as shown in the table III-B, VII, and VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' FRCN along with synthetic data gives the best mAP accuracy on VOC 2007 dataset as shown in Table VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Several classical and automatic data augmentation methods have shown the promising performance on different state-of-the-art models on PASCAL VOC dataset as shown in table III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The DetAdvProp achieves the highest score on every model and mAP, AP50 and AP75 metrics on PASCAL VOC 2012 dataset, outperforming AutoAugment [23] as shown in the table VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The performance is reported in average precision (AP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' AP50 and AP75 are the average precision with 50% and 75% threshold, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Semantic Segmentation This subsection includes semantic segmentation results on PASCAL VOC and CITYSCAPES datasets, most frequently used in several research papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' In the table (IX) and table (X), we compiled the effectiveness of validation set results on the different datasets (mIoU) with data augmentation on semantic segmentation models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' the best results of performance (mIoU) accuracy on the Cityscape dataset as shown in ta- ble (ix) and best results of performance (mIoU) accuracy on Pascal VOC datasets are shown in table (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' We found performance gains on a few metrics with several semantic segmentation models: deeplabv3+ [144], DeepLab-v2 [93], Xception-65 [144], ExFuse [150] and Eff-L2 [156] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' All semantic segmentation models have been found to perform better when data augmentation techniques are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Traditional data augmentation methods are rotation, scaling, flipping and shifting [148].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' DISCUSSION AND FUTURE DIRECTIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Current approaches It is proven that if we provide more data to the model, it improves model performance [43], [121].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' A few current tendencies are discussed by Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' [141].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Among these, one way is to collect the data and label it manually, but it is not an efficient way to do this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Another efficient way is to apply data augmentation, the more data augmentations we apply, the more performance improves to a certain extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Currently, image mixing methods and autoaugment methods are successful for image classification tasks, scale aware based auto augment methods are showing promising results in detection tasks and semantic segmentation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' But these data augmentation performances can vary with the number of data augmentation applied, as it is known that the combined data augmentation methods show better performance than single one [97], [142].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Theoretical aspects There is no theoretical support available to explain why specific augmentation is improving performance and which sample(s) should be augmented, as the same aspect has been discussed by Yang et al [142] and Shorten et al [108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Like in random erasing, we randomly erase the region of the image - sometime may erase discriminating features, and the erased image makes no sense to a human.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' But the reason behind performance improvement is still unknown, which is another open challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Most of the time, we find the optimal parameters of the augmentation through an extensive number of experiments or we choose data augmentation based on our experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' But there should be a mechanism for choosing the data augmentation with theoretical support considering model architecture and dataset size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Researching the theoretical as- pect is another challenge open for the research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Optimal number of samples generation It is a known fact, as we increase data size, it improves the performance [43], [108], [121], [142] but it is not a case - increasing the number of samples will not improve performance after a certain number of samples [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' What is the optimal number of samples to be generated, depending on the model architecture and dataset size, is another aspect to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Currently, researchers perform many experiments to find the optimal number of sample generation [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' But it is not feasible way as it requires time and computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Can we devise a mechanism to find an optimal number of samples, which is an open research challenge?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Selection of data augmentation based on model archi- tecture and dataset Data augmentation selection depends on the nature of the dataset and model architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Like on MNIST [25] dataset, geometric transformations are not safe such as rotation on 6 and 9 digits will no longer preserve the label information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' For densely parameterized CNN, it is easy to overfit on weakly augmented datasets, and for shallow parameterized CNN, it may break generalization capability with data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It suggests, while selecting the data augmentation, the nature of the dataset and model architecture should be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It is not an easy problem to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Currently, numer- ous experiments are performed to find model architecture and suitable data augmentation for a specific dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Devising a systematic approach to select the data augmentation based on dataset and model architecture is another open challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Augmentations for spaces Most of the data augmentation has been explored on the image level - data space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Very few research works have explored data on feature level - feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Challenges here arise, in which space should we apply data augmentation, data TABLE III COMPARISON ON CIFAR-10 AND SVHN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' NUMBER REPRESENTS ERROR RATES ACROSS THREE RUNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' CIFAR-10 SVHN Method 40 labels 250 labels 1,000 labels 4,000 labels 40 labels 250 labels 1,000 labels 4,000 labels VAT [91] 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='03 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='82 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='40 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='31 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='41 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='01 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='15 Mean Teacher [123] 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='32 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='71 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='32±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='00 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='36±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='45±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='43 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='75±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='39±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='11 MixMatch [9] 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='54±11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='50 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='08±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='87 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='75±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='32 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='24±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='06 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='55±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='53 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='78±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='27±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='31 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='89±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='06 ReMixMatch [8] 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='10±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='64 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='27±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='34 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='73±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='16 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='14±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='34±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='10±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='83±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='42±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='09 UDA 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='05±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='93 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='76± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='90 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='87± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='29± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='25 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='63±20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='51 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='76± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='55± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='09 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='47± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='15 SSL with Memory [17] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='22 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='83 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='21 Deep Co-Training [99] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='35± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='06 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='29 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='03 Weight Averaging [5] 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='58 I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='12 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='05± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='21 ICT [126] 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='48 I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='78 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='29± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='02 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='78 I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='68 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='89 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='04 Label Propagation [57] 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='70 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='28 SNTG [87] 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='52 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='89 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='34 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='29± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='86 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='27 PLCB [4] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='85 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='97± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='15 II-model [105] 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='02 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='05 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='98 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='41± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='37 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='65 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='27 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='60± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='18 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='57± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='14 PseudoLabel [77] 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='98 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='17 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='91 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='73 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='11 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='16± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='88 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='41 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='71± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='07 Mixup [147] 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='92 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='66 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='20 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='97 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='89 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='63 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='96 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='14 FeatMatch [73] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='50 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='64 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='76 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='07 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='91± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='34± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='10± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='06 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='02 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='37±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='43 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='49±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='07 TABLE IV COMPARISON ON CIFAR-100 AND MINI-IMAGENET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' NUMBER REPRESENTS ERROR RATES ACROSS TWO RUNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' CIFAR-100 mini-ImageNet Method 400 labels 4,000 labels 10,000 labels 4,000 labels 10,000 labels II-model [105] 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='19± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='36 SNTG [87] 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='97± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='29 SSL with Memory [17] 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='51± 0.' metadata={'source': 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+page_content='49 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='08± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='51 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='51± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='22 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='55 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='11 Label Propagation [57] 43.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='11 FeatMatch 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='41 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='04 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='06 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='79±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='22 MixMatch 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='61±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='32 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='31±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='33 UDA 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='28±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='88 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='50±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='25 ReMixMatch 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='28±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='06 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='03±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='56 FixMatch 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='85±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='75 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='60±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='12 space or feature space?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It is another interesting aspect that can be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' For approaches, it seems it depends on the dataset, model architecture and task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Mixing augmentations in feature space is senseless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Currently, approaches are conducting ex- periments in data space and feature space and then selecting the best one [138].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' This is not the optimal way to go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It is still an open challenge to be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Open research questions Despite the success of data augmentation techniques in dif- ferent CV tasks, it still failed to solve challenges in SOTA data augmentation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' After thoroughly reviewing SOTA data augmentation approaches, we found several challenges and difficulties, which are yet to be solved, as it is listed below: In image mixing techniques, label smoothing has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' It makes sense whatever portion of images is mixed, corresponding labels should be mixed accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' To the best of our knowledge, none has explored label smoothing for image manipulation and image erasing subcategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' For example, if the image portion is randomly cut out in cutout data augmentation, the corresponding label should be smoothened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' The same rule applies to the image erasing category and image manipulation - where the image part is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Currently, data augmentation is performed without con- sidering the importance of an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' All examples may not be difficult for the neural network to learn, but some are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' So augmentation should be applied to those difficult examples by measuring the importance of the examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' In image mixing data augmentations, if we mix more than two images salient parts, that are truly participating in augmentation unlike RICAP [122], what is its effect?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Note, the corresponding labels of these images will be mixed accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' In random data augmentation under auto augmentations, the order of augmentations has not been explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' We be- lieve it has significant importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' What are the possible ways to explore the order of existing augmentations such TABLE V COMPARISON OF TEST ERROR RATES ON CIFAR-10 & SVHN USING WIDERESNET-28 AND CNN-13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Approach Method CIFAR-10 (Nl=4000) SVHN(Nl=1000) WideResNet-28 Supervised 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='38 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='47 Pseudo PL [77] 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='57 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='29 Labeling PL-CB [4] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='3 II Model [75] 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='63 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='27 Mean Teacher [123] 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='28 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='65 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='47 VAT [91] 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='27 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='20 Consistency VAT + EntMin [91] 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='13 I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='39 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='35 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='19 LGA + VAT [58] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='19 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='36 Regularization ICT [126] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='07 MixMatch [9] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='06 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='31 UDA 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='17 ReMixMatch (Berthelot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' 2020) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='14 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='09 FixMatch [115] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='11 CL 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='03 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='11 Pseudo CL+FA [82] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='19 Labeling CL+FA [82]+Mixup [147] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='15 CL+RA+Mixup [147] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='188 CNN-13 Pseudo Labeling TSSDL-MT 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='55 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='27 LP-MT 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='61±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='28 Ladder net [102] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='36±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='31 MeanTeacher [123] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='24 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='19 Temporal ensembling [75] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='16 Consistency VAT [91] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='34 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='42 Regularization NATEntMin [91] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='86 SNTG [87] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='27 ICT [126] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='04 Pseudo CL 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='22 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='28 Labeling CL+RA 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='07 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='10 as first traditional data augmentations and then image mixing or weight-based?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' If we mix the masks of the objects in data augmentation for semantic segmentation, How does the model behave and what is its effect?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Finding the optimal ordered number of data augmentation and the optimal number of samples to be augmented is another open challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' For example, in randAug method there are N optimal number of augmentations was found but it is not known how many samples should be augmented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' CONCLUSION This survey presents numerous SOTA data augmentation methods to cope with overfitting problems in computer vision tasks due to data limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' We provided a comprehensive survey for data augmentation, in which we presented novel taxonomy of advanced data augmentation approaches, an overview of each SOTA data augmentation, and results of numerous computer vision tasks such as image classifica- tion, object detection and semantic segmentation, with data augmentation effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' We not only compiled the results for supervised learning tasks but also compiled results for semi- supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' For result reproducibility, we compiled the available codes of the data augmentation by following the proposed taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' We discuss a different aspects of the data augmentation with its difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Finally, we discuss the open research questions, which are very promising and open new doors, and ignite interest in the research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' We believe that the survey benefits the researchers as follows: i) Understanding of the data augmentation ii) No need to find the results for comparison purposes iii) Results can be reproduced with available codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' ACKNOWLEDGMENT This publication has emanated from research [conducted with the financial support of supported in part by a grant from] Science Foundation Ireland under Grant number 18/CRT/6223 and is supported by the ADAPT Centre for Digital Content Technology which is funded under the SFI Research Cen- tres Programme (Grant 13/RC/2106/P2), Lero SFI Centre for Software (Grant 13/RC/2094/P2) and is co-funded under the European Regional Development Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' For the purpose of Open Access, the author has applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission REFERENCES [1] Jiwoon Ahn, Sunghyun Cho, and Suha Kwak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Weakly supervised learning of instance segmentation with inter-pixel relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' In Pro- ceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 2209–2218, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Method Detector BackBone AP AP50 AP75 APs APm APl Hand-crafted: Dropblock [38] RetinaNet ResNet-50 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='4 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='4 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='2 − − − AutoAugment+color Ops [155] RetinaNet ResNet-50 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='5 − − − geometric Ops [155] RetinaNet ResNet-50 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6 − − − bbox-only Ops [155] RetinaNet ResNet-50 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='0 − − − Mix-up [151] Faster R-CNN ResNet-101 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='1 PSIS* [128] Faster R-CNN ResNet-101 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='2 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='1 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='2 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='3 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='7 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6 Stitcher [19] Faster R-CNN ResNet-101 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='1 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='9 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='5 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='1 GridMask [15] Faster R-CNN ResNeXt-101 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='0 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='5 InstaBoost* [32] Mask R-CNN ResNet-101 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='0 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='3 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='2 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='9 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6 SNIP (MS test)* [112] Faster R-CNN ResNet-101-DCN-C4 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='4 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='2 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='9 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='3 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='4 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='9 SNIPER (MS test)* [113] Faster R-CNN ResNet-101-DCN-C4 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='1 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='0 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='9 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='1 Traditional Aug [142] Faster R-CNN ResNet-101 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='80 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='0 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='0 Traditional Aug* [29] CenterNet ResNet-101 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='15 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='01 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='30 Traditional Aug+ [15] Faster-RCNN 50-FPN (2×) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='4 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='7 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='5 Traditional Aug+ [15] Faster-RCNN 50-FPN (2×)+GridMask (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='3) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='2 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='0 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='4 Traditional Aug+ [15] Faster-RCNN 50-FPN (2×)+ GridMask (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='5) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} 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Traditional Aug+ [15] Faster-RCNN X101-FPN (2×)) 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='4 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='2 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8 Traditional Aug+ [15] Faster-RCNN X101-FPN (2×)+ GridMask (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='7)) 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='0 46.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='2 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='3 DADAAugment: [80] Mask R-CNN ResNet-50(DADA) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='1 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='4) + Det-AdvProp 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='2 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8) 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='7) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='7 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='9) TABLE VIII 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='87 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='7 DST-CBC [34] DeepLabV2 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='5 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='4 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='9 ClassMix [93] DeepLabV2 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='4 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='1 s4GAN [90] DeepLab-v2 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='3 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='2 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='4 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6 French et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='el [35] DeepLab-v2 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='79 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='81 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='48 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='60 DST-CBC [34] DeepLab-v2 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='5 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='3 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='7 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8 ClassMix:Seg* [93] DeepLab-v2 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='18 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='15 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='77 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='00 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='45 Mixup [147] IRNet 49 CutOut [27] IRNet 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='9 CutMix [146] IRNet 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='2 Random pasting [119] IRNet 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8 CCNN [96] VGG16 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6 SEC [66] VGG16 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='1 STC [135] VGG16 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='2 AdvEra [134] VGG16 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='7 DCSP [13] ResNet101 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='9 MDC [136] VGG16 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8 MCOF [131] ResNet101 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='2 DSRG [54] ResNet101 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='2 AffinityNet [2] ResNet-38 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='7 IRNet [1] ResNet50 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8 FickleNet [78] ResNet101 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='3 SEAM [133] ResNet38 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='7 ICD [31] ResNet101 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='3 IRNet + CDA [119] ResNet50 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='4 SEAM + CDA [119] ResNet38 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8 DeepLab V3 [148] MobileNet 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='9 DeepLab V3 [148] ResNet-50 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8 DeepLab V3 [148] ResNet-101 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='4 DeepLab V3plus [148] MobileNet 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8 DeepLab V3plus [148] ResNet-50 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8 DeepLab V3plus [148] ResNet-101 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6 Baseline+R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Rotation [148] ObjectAug 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='5 Baseline +R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Scaling [148] ObjectAug 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='3 Baseline + R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Flipping [148] ObjectAug 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6 Baseline + R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='Shifting [148] ObjectAug 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='7 Baseline + All [148] ObjectAug 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8 Baseline + CutOut (16×16, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='5) [148] MobileNet 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='9 Baseline + CutOut (16×16, p = 1) [148] MobileNet 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='3 Baseline + CutMix (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='5) [148] MobileNet 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='7 Baseline + CutMix (p = 1) [148] MobileNet 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='4 Baseline + ObjectAug [148] MobileNet 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8 Baseline + CutOut (16×16, p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='5) + ObjectAug [148] MobileNet 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='9 Baseline + CutMix (p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='5) + ObjectAug [148] MobileNet 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='1 DeepLabv3+ [14] EfficientNet-B7 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='6 ExFuse [150] EfficientNet-B7 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='8 Eff-B7 [156] EfficientNet-B7 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='2 Eff-L2 [156] EfficientNet-B7 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='7 Eff-B7 NAS-FPN 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In Proceedings of the AAAI conference on artificial intelligence, volume 34, pages 13001–13008, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' [155] Barret Zoph, Ekin D Cubuk, Golnaz Ghiasi, Tsung-Yi Lin, Jonathon Shlens, and Quoc V Le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Learning data augmentation strategies for object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' In European conference on computer vision, pages 566–583.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Springer, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' [156] Barret Zoph, Golnaz Ghiasi, Tsung-Yi Lin, Yin Cui, Hanxiao Liu, Ekin Dogus Cubuk, and Quoc Le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Rethinking pre-training and self- training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Advances in neural information processing systems, 33:3833– 3845, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' [157] Yuliang Zou, Zizhao Zhang, Han Zhang, Chun-Liang Li, Xiao Bian, Jia-Bin Huang, and Tomas Pfister.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' Pseudoseg: Designing pseudo labels for semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content=' arXiv preprint arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} +page_content='09713, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQf_wJK/content/2301.02830v1.pdf'} diff --git a/39E4T4oBgHgl3EQf0g2Q/content/tmp_files/2301.05283v1.pdf.txt b/39E4T4oBgHgl3EQf0g2Q/content/tmp_files/2301.05283v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4db6b6f3d0ab31d77618061682fdac971ed57a23 --- /dev/null +++ b/39E4T4oBgHgl3EQf0g2Q/content/tmp_files/2301.05283v1.pdf.txt @@ -0,0 +1,1397 @@ +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: ikozlov90@gmail.com +†Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991 Russia, E-mail: +a@oshemkov.ru +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 𝑑𝑘± +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 + diff --git a/39E4T4oBgHgl3EQf0g2Q/content/tmp_files/load_file.txt b/39E4T4oBgHgl3EQf0g2Q/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a45c2a68eb61224e15e7ae89ecbd3a1778759d79 --- /dev/null +++ b/39E4T4oBgHgl3EQf0g2Q/content/tmp_files/load_file.txt @@ -0,0 +1,352 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf,len=351 +page_content='Integrable systems with linear periodic integral for the Lie algebra e(3) I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'} +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′ 3𝑅3.' 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'} +page_content=' References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'} +page_content=' Bolsinov and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'} +page_content='T.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'} +page_content=' 19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQf0g2Q/content/2301.05283v1.pdf'} diff --git a/6NAyT4oBgHgl3EQfcffh/content/tmp_files/2301.00286v1.pdf.txt b/6NAyT4oBgHgl3EQfcffh/content/tmp_files/2301.00286v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..53845e03d950c76ae70f65367e6c976dba27220c --- /dev/null +++ b/6NAyT4oBgHgl3EQfcffh/content/tmp_files/2301.00286v1.pdf.txt @@ -0,0 +1,469 @@ +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 + diff --git a/6NAyT4oBgHgl3EQfcffh/content/tmp_files/load_file.txt b/6NAyT4oBgHgl3EQfcffh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..04efcaf790a888306e7c2191391345664c7a4504 --- /dev/null +++ b/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'} +page_content=' Jackson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'} +page_content=' Infinite families of bi-embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'} +page_content=' Discrete Mathematics, 82(2):127–141, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'} +page_content=' [Get18] Ellen Gethner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'} +page_content=' To the Moon and Beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'} +page_content=' In Graph Theory—Favorite Conjectures and Open Problems – 2, pages 115–133.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'} +page_content=' 8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfcffh/content/2301.00286v1.pdf'} diff --git a/6dAyT4oBgHgl3EQfpfjS/vector_store/index.faiss b/6dAyT4oBgHgl3EQfpfjS/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..49aa95a3c4ff12edaae80af26213c9914a448b8d --- /dev/null +++ b/6dAyT4oBgHgl3EQfpfjS/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:db49099937469309d8d1f3c84ec806c4de4d684017105688117b85ee8f6d2795 +size 1441837 diff --git a/8tE4T4oBgHgl3EQf3A1I/content/tmp_files/2301.05302v1.pdf.txt b/8tE4T4oBgHgl3EQf3A1I/content/tmp_files/2301.05302v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c5959cc5059f12d0c109e53d7b09593d67c05bf9 --- /dev/null +++ b/8tE4T4oBgHgl3EQf3A1I/content/tmp_files/2301.05302v1.pdf.txt @@ -0,0 +1,2324 @@ +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]. 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Nistor, Nonlin- +ear normal modes of a two degrees-of-freedom piecewise +linear system, Mechanical Systems and Signal Processing +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. + diff --git a/8tE4T4oBgHgl3EQf3A1I/content/tmp_files/load_file.txt b/8tE4T4oBgHgl3EQf3A1I/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e512d8056bed0d431d5d7843d73f985f3fc546d8 --- /dev/null +++ b/8tE4T4oBgHgl3EQf3A1I/content/tmp_files/load_file.txt @@ -0,0 +1,1199 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf,len=1198 +page_content='Wavenumber Scattering and Inter-band Targeted Energy Transfer in Phononic Lattices with Local Vibro-Impact Nonlinearities Joshua R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Tempelman, Alexander F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Vakakis, and Kathryn H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' this power law dependence is additionally validated by the ROM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' INTRODUCTION Periodicity has been leveraged to control acoustic and elastic energy propagation in linear time-invariant (LTI) phononic metamaterials [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Whereas band manipulation has been achieved by in- troducing e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=', 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='05302v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='app-ph] 12 Jan 2023 2 mitigation [38–52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The use of nonlinear attachments in acoustic wave guides (either bulk or periodic) have demonstrated un- precedented properties in acoustical systems [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Moreover, with the incorporation of hierarchical mass scales and asym- metry, similar systems have achieved nonreciprocity [60– 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Unlike resonant mechanisms, non-resonant mechanisms aim to scatter energy across the underlying linear modal basis in a low-to-high frequency fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' In a similar fashion, Theurich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' To date, non-resonant energy scattering concepts have not been extended to periodic phononic metamaterials from a wave propagation perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Hence, several natural questions arise when considering non-resonant TET phenomena in a phononic material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' This paper addresses these questions with extensive computational probing, new post-processing techniques, and physics- based reasoning of the resulting nonlinear acoustic phe- nomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Next, we consider a 4-band phononic lattice, which has one acoustic and three opti- cal bands over a relatively broad frequency/wavenumber range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' This brings about the new non- linear acoustic phenomenon of inter-band targeted energy transfer (IBTET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Accordingly, the organization of this paper is as fol- lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Lastly, Section V of- fers concluding remarks and some suggestions for further extension of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The system is compu- tationally explored by performing numerical simulations with wave packet excitations over an array of excitation amplitudes and wave numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The nominal unit cell (b) without a VI nonlinearity and (c) with the VI nonlinearity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' (d) schematics of finite lattice con- figurations which are comprised of the linear phononic lattice with various number of embedded VI unit cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' This yields two pass bands for this lattice, namely a lower-frequency acoustic band and a higher- frequency optical band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The nonlinear VI cells that are locally distributed through the lattice provide the following VI forces,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' FNL(wk) = kc � (wk − ∆i)n + − (−wk − ∆k)n + � g( ˙wk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' ˙w− k ) (4) where wk(t) = uk 2(t) − uk 1(t) is the relative deflection be- tween the resonator and its host mass,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' n the nonlinearity coefficient which is set to n = 3/2 to emulate Hertzian contact unless otherwise stated,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' ∆k the clearance of the k-th VI in the lattice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' and kc = 2EVI √RVI 3(1−ν2) the stiffness parameter for Hertzian contacts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' with EVI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' RVI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' and ν being the modulus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' radius,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' and Poisson ratio of the VI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The notation ( )+ indicates that only pos- itive arguments are to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Note that Eq (4) does not modify the underling linear band structure of the extended lat- tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Moreover, for amplitudes such that wk < ∆k for each VI, the wave propagation remains completely linear as no VI experiences contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Numerical simulations of equations (2) were carried out using the ODE78 routine in MATLAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' however we focus only on κ⋆ = 5π/9 and refer the reader to supplemental material for additional results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Clearances were nominally set to range between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='0002 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The mass and stiffness of the linear resonator (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=', in the absence of rigid barriers and VIs - cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 1) were selected to em- ulate realistic resonator systems considered in the liter- ature [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Table I lists nominal parameters for stiffness, mass, and VI stiffness parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Within this frame- work, an ensemble of simulation data was constructed for 25 logarithmically increasing forcing amplitudes for each 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' configuration and excitation wavenumber considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='08 90 90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='005 200 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Quantitative results across FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The spatial wavelet transformations of the propagat- ing waves considered in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 2 for (a) low, (b) medium, and (c) high excitation amplitude;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' all simulations will be given subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 2 depicts the results for a representative simu- lation with a 5-VI configuration (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 1) for low, medium, and high forcing amplitude (equivalently low, medium, and high energy simulations) corresponding A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='1, 1, and 10 N, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' At low amplitude, the acoustics are entirely linear as the wave does not create deflections greater than the VI clearance (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 2(ai)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 2 (bi,ci)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' To this end, we utilize a set of signal processing procedures which are briefly detailed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 2(aii)-(cii) depict the wavenumber distributions across the lattice computed over progres- sions of time snap shots for each simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' It is clear from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' In contrast, new wave numbers emerge for medium and high excitation amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 2(ci) do not generate substantial wavenumber components beyond that of the incident wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 2(aiii)- (ciii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Note that Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 2(aiii)-(ciii) consider the en- tire time record of the simulation from start to finish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 2(i,ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Note that the spectral content generated by scattering in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 2(biii) remains bound to the underlying linear dispersion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' this acoustical nonlinear scattering effect is directly equivalent to the nonresonant scattering mech- anisms studied in modal dynamics [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Information regarding the spatial evolution of the gen- erated wavenumber components over space and time re- quires a space-frequency analysis routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 2(ii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Lastly, it is apparent from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 3(b) that certain wavenumber components propa- gate much faster than others and all follow behind the incident wavenumber;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' this is a direct consequence of FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' (b) superimposition of wave propagation at each wavenumber partition depicted by contours for (i) low, (ii) medium, and (iii) high energy simulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' (c) the optical band of the linear lattice plotted with corresponding colors to the wavenumber-based energy contours of (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' the dispersion relation of the underlying linear system (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 1) which is steepest towards the center of the optical band and therefore corresponds to larger group velocity at the incident wavenumber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' However, the general trends of spectral gener- ation with respect to energy are consistent nevertheless (see supplemental information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The spectral content of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 3 can be mapped-back into the spatial-temporal domain by considering a spec- tral partitioning scheme similar to that presented in [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' K12(x, t) = � j W−1(X(κ, x)) �� t=tj, 11π 12 ≤ κ ≤ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 4 depicts the results of the wavenumber parti- tioning scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' To demonstrate the dependency on energy, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 4(c) provides a colored depiction of the optical band to make the contours of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 4(b) more obvious with respect to which wavenumber components are generated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' the of group velocities in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 4(c) corre- sponds directly to the variable wave speeds of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 4(b), and this can be used to interpret the variation in spatial- spectral propagation of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 3 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' To this point, the results have FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Mean spectral entropy in the lattice with VIs for system configurations ranging between 1 VI to 20 VI (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 1) over an array of excitation amplitudes logarithimcally spaced from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='1 to 20: Top and bottom plots are for the same data with the bottom plots depicting the log-log scaling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' been presented in a largely qualitative manner with an emphasis on graphical interpretations (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 2,3,4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' To this end, we make use of information theory by considering the spectral entropy of the nonlinear acous- tics in the wavenumber domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Spectral entropy is the extension of classical Shannon entropy to the frequency domain [75] and is a standard metric for quantifying sig- nal complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 5 depicts the normalized average entropy quantity with respect to forcing amplitude for all configurations depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Normalization was performed so that the minimum and maximum entropy for each VI configuration range be- tween 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='01 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The log-log plots of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='95 being recovered for the majority of configurations studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Error bars in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 5 measure the standard deviation of entropy across the spatial extent of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' This can be interpreted as a measure of how uniform the wavenumber complexity is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 2, 3, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Note that the excitation wavenumber is κ = 5π/9 for all results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' additional results given in the supplemental material confirms that the same trends hold across all incident wavenumbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' This can be considered as an acoustics- equivalent to the IMTET nonlinear mechanism estab- lished in dynamics [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Hence, to maintain the simplicity of 1D, we proceed with a 4-DoF model of the unit cell, offering (a) (b) UNIT CELL FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Increasing the bands of the lattice: (a) Schematic of the unit cell, and (b) the corresponding dispersion diagram for parameters λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='1 and η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' two additional bands to transfer energy towards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The 4-band Lattice The 4-band model emulates closely the resonator model of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The main difference is that masses have been added in-series in between resonators as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' IBTET in the 4-band lattice with 5 VI sites: (a) shows the evolution of the propagating wave energy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' (b-e) propagation of the wave energy corresponding to each band of the lattice based on the numerically recovered dispersion of the full simulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' (f,g) dispersion of the input and output segments (labeled in (a)) demonstrating the targeted energy transfer to the higher bands;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' ˜u2(κ)˜u3(κ) < 0 ∀κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' (8) We confine this search for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='1 < λ < 1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='1 < η < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' With this constraint, minimizing the cost function over (λ, η) is trivial and returns λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='1 and η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The resulting band structure is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Accordingly, we consider only a 5-VI lattice configura- tion (as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 1(d)) herein and refer the reader to supplemental material for the results of a 1-VI lattice configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Simulations were performed similarly to section II with excitation provided by a windowed tone burst (Eq (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Simu- lations were performed for 50 selections of the excitation amplitude between 1 and 104 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' For the 4- band system considered, the contact stiffness parameters (kc) were computed based on E = 100 MPa, ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='3, and RVI = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='005 m, and the clearances are now varied between 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='65 and 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='75 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Low-to-high band targeted energy transfer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 7 depicts an example of a wave propagating through the 4-band system with five Herzian VIs en- gaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Energy clearly cascades from the main wave packet as it propagates through the lattice (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 7(a)), similar to the diatomic chain (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 7(f,g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' This is further confirmed by Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' hence the low-to-high frequency targeted transfer of en- ergy from the second band to the higher bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' There- after, the 2-D spectrum is partitioned band-by-band and also into band-gap regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' This results in the propagation depicted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Note that these low, medium, and high excitation amplitudes now refer to order 1, order 10, and order 100 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' To verify and quantify the efficacy of the VIs to induce TET from low- to-high bands (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=', 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 9 depicts the results of the IBTET analysis over the ranges of forcing amplitudes considered for both Hertzian and bilinear VI laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' This is on the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' This is further verified in the linearly-scaled plots of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 9(aii,bii) which show that the maximum energy transferred to the higher optical bands is roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='3-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='35 (30-35%) for both the Hertzian and bilinear VIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' A 2-DoF model emulating a VI resonator cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' metamaterials that could exhibit IBTET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Such methodology has been employed already for under- standing the dynamical evolution of VI systems of various forms [71, 80, 81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' (9) where an overbar denotes that the variable is associated with the ROM and not the full phononic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='75 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Nonlinear Normal Modes as a Measure of Nonlinearity The energy dependencies of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 5 and 9 make an NNM approach a natural avenue since continuation re- turns an overview of the dynamics across energy scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' We provide a grossly condensed description herein and refer the reader to [79] for full algorithmic details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The state form of sys- tem (9) is ˙z = g(z) where g(z) is a nonlinear function of the state variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Newton’s method can be used to recover periodic solu- tions at low energy in the shooting stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' We define the phase condition such that the two DoFs of the ROM have zero initial velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' After shooting is completed, a pseudo-arclength method is used to trace out the NNM branch in the 2n + 1 dimensional parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 11 depicts the FEPs computed for system described by equation (9) for both Hertzian and bilinear contact laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The FEP results reveal similar trends for both Hertzian 11 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' (c,d) slopes of the FEPs of of (a,b) with respect to energy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' and bilinear VI ROMS, possessing four dynamical region labeled (I)-(IV) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The corresponding phase tra- jectories of the periodic orbits in each region are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 11(e) and 11(f) for Hertzian and Bilinaer mod- els, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' In the low energy region (I), the VIs do not engage, and the dynamics are completely linear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' this is confirmed by zero slope of the FEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' This in turn yields a small neighborhood of the NNM branch where the FEP slope is theoretically infi- nite, and the subplots of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 11(c,d) confirm that this is where to maximum is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The phase trajectories indicate that region II represents a transition where the dynamics are most sensitive to nonlinear effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Despite the apparent smoothness of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 11(eII,fII) the volatile VI-grazing dynamics in region II are unstable, and hence, not physically realizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 11 for portions of the NNM branch with Floquet multiplier, α, far exceeding 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' After the grazing VI region in region II is surpassed with increasing energy, the FEP gradually increases in frequency towards region III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Region III is character- ized by strong VI oscillations which is apparent by the box-like phase trajectories indicating non-smooth tem- poral dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' In this region, the linear dynamics of ˆk1 are negligible and the VI dynamics dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Region IV mani- fests smooth dynamics characterized by in-phase dynam- ics predominantly dictated the contact stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' In this region, the clearance is negligible and the VI contacts be- have as an extremely stiff elastic spring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 11(eIV,fIV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Moreover, for the bilinear system, the FEP clearly levels off as the high-energy (al- most) linear limiting behavior is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 11(b,c)) posses a remarkable similar- ity to the observed trends of nonlinear IBTET in the full phononic lattice (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The two measures can be related to one another by replotting the energy trans- fers of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' These normalizations are performed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The FEP 12 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The relative inter-band energy transfer, with the normalized slope from the ROM-FEP superimposed for (a) Hertzian and (b) bilinear contact models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' A scalar quantity defined by the low-bound of IBTET (dashed lines of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 9) is then added to the FEP slope account for the lower threshold of the energy transfer in the VI lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' These normaliza- tions preserve the slopes of both quantities since scalar multiplication results only in translations in log scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Hence, the previous measures can be directly compared with respect to their decrease in value with respect to increasing normalized energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Hence, the underlying FEP of the ROM, along with the evolution of the dynamical regimes of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 11, clearly have a direct implication of the IBTET in the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' CONCLUSIONS In this work, we have investigated the effect of local VI nonlinearities on the propagation of traveling waves in 1-D phononic lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Specifically, first a di-atomic 2-band lattice was numerically studied over a wide range of forcing amplitudes and embedded VI configurations (section II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Simula- tions were carried out over a range of excitation ampli- tudes with both Hertzian and bilinear contact laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Nu- merical post-processing reconstructed the energy of each band, and it was shown that IBTET is indeed possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Although this work focused primarily on fundamental understanding of the physics at play, the implications and potential for future developments are rather extensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The low-to-high energy transfers directly correspond to a reduction in magnitude, since the energy must be pre- served in the frequency transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' These attributes alone make VI-based methods attractive for wave transmission tuning (or tailoring) with respect to amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported in part by the National Sci- ence Foundation Graduate Research Fellowship Program under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' DGE – 1746047.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' DETAILS ON SIGNAL PROCESSING PROCEDURES 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' A similar discourse may be found in [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The CWT is traditionally used as a time-frequency analysis tool by transforming the signal from the time domain to the time-frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' To the same effect, one can consider the space-wavenumber domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' (11) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The reconstructed kinetic energy and correspond- ing reconstruction error for the described wavelet partition scheme;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' red dashed line indicates 1 percent error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' We consider the Morelet wavelet for all transformations in this work: ψ(x) = 1 π1/4 � eiκcx − e−κ2 c/2� e−x2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' (12) For the scale and quantities of datasets considered in this work, computational efficiency is a requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' To this end, the Fast Fourier Transform is employed to speed up wavelet computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' (14) Each wavelet transformation can be partitioned over space and wavenumber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' (16) where ˆXk(κ, ξ) is the Fourier transformation of Xk(κ, x) with respect to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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)|| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' (17) 14 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Spectral Entropy Here, we provide more details pertaining to the spec- tral entropy plots displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 14 depicts the distribution of entropy using Eq (6) to recover H(x) for each t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The resulting matrix H(x, t) is plotted as an image for low, medium, and high excitation amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The distribution of high-entropy regions is clearly seen in the medium and high excitation amplitude simulations as the VIs engage the incoming wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Superimposed on each image is the instantaneous spectral entropy, which summarizes H(x, t) over space to render time-dependent measures H(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' This is depicted in the bottom plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' In the low- amplitude region with no VI engagement, no entropy is generated after excitation (as expected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' For medium amplitudes, regions of sustained high wavenumber en- tropy are realized after the VIs engage the incident wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Energy Reconstruction of band-partitioning decom- position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Computing energy on each band The computation of wave energy over each band in section III is performed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' frequency filters are constructed as follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Gk(κ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' ω) = � 1 ω ∈ Bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' −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),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' and of for the band gap between the upper two optical bands (BG2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' B5 = min(BG1) ≤ ω ≤ max(BG1) B6 = min(BG2) ≤ ω ≤ max(BG2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' NONLINEAR NORMAL MODE COMPUTATIONS The recipe for NNM calculations follows very closely to the procedure outlined in [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' In region II, the unstable dynamics proved to be challenging for the computation of the corre- sponding NNM branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Hence, sufficiently small predic- tor steps were required for convergence, with the residual reduction being varied from 10−12 to 10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Sensitivity analysis was employed again to compute Jacobian terms in region II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Mikhlin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Pilipchuk, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Zevin, Normal Modes and Localiza- tion in Nonlinear Systems (Wiley & Sons, Incorporated, John, 2008) p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 552.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' [77] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Kerschen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Peeters, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Golinval, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Vakakis, Nonlinear normal modes, part i: A useful framework for the structural dynamicist, Mechanical Systems and Sig- nal Processing 23, 170 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' [78] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Avramov and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Mikhlin, Review of applications of nonlinear normal modes for vibrating mechanical sys- tems, Applied Mechanics Reviews 65, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='1115/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='4023533 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' [79] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Peeters, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Vigui´e, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' S´erandour, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Kerschen, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Golinval, Nonlinear normal modes, part II: To- ward a practical computation using numerical continu- ation techniques, Mechanical Systems and Signal Pro- cessing 23, 195 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' [80] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Tao and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Gibert, Periodic orbits of a conservative 2-DOF vibro-impact system by piecewise continuation: bifurcations and fractals, Nonlinear Dynamics 95, 2963 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' [81] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Moussi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Bellizzi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Cochelin, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Nistor, Nonlin- ear normal modes of a two degrees-of-freedom piecewise linear system, Mechanical Systems and Signal Processing 64-65, 266 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 1 Supplemental Materials: Wavenumber Scattering and Inter-band Targeted Energy Transfer in Phononic Lattices with Local Vibro-Impact Nonlinearities Joshua R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Tempelman, Alexander F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Vakakis, Kathryn H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Matlack Department of Mechanical Science and Engineering, University of Illinois at Urbana Champaign 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' ADDITIONAL INFORMATION FOR WAVENUMBER SCATTERING Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' S1 provides a graphical illustration of the signal processing processes described in section II and Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Starting in the spatio-temporal domain, snap-shots of the wave velocity are taken successively and converted into the wavelet domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' This domain is partitioned into 12 bands (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' S1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The inverse transformation of the k-th band partition gives at a fixed point in time gives the velocity vector ˙uk(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The energies are contacted over time to deliver the energy corresponding to wave propagation on the k-th band;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Graphical illustration of the wavelet-based wavenumber partitioning processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' K12 3 K11 K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='5 K10 Kg Wavenumber 2 K: K7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='5 K6 K5 K4 K10 K4 K: K5 K11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='5 K2 Ki 0 K6 K12 100 200 300 400 500 600 Unit Cell No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='Lotal3Energies Summary (Transformed Coords 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='5 Reconstruced Kinetic Energy Physical Kinetic Energy Energy of Individual Partitions 2 60 nerg 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='5 K10 Kg Wavenumber 2 K: K7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='5 K6 K5 K4 K10 K4 K: K5 K11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='5 K2 Ki 0 K6 K12 100 200 300 400 500 600 Unit Cell No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='K12 3 K11 K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='5 K10 Kg Wavenumber 2 K: K7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='5 K6 K5 K4 K10 K4 K: K5 K11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='5 K2 Ki 0 K6 K12 100 200 300 400 500 600 Unit Cell No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='K12 3 K11 K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='5 K10 Kg Wavenumber 2 K: K7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='5 K6 K5 K4 K10 K4 K: K5 K11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='5 K2 Ki 0 K6 K12 100 200 300 400 500 600 Unit Cell No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='K12 3 K11 K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='5 K10 Kg Wavenumber 2 K: K7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='5 K6 K5 K4 K10 K4 K: K5 K11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='5 K2 Ki 0 K6 K12 100 200 300 400 500 600 Unit Cell No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='K12 3 K11 K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='5 K10 Kg Wavenumber 2 K: K7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='5 K6 K5 K4 K10 K4 K: K5 K11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='5 K2 Ki 0 K6 K12 100 200 300 400 500 600 Unit Cell No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='K12 3 K11 K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='5 K10 Kg Wavenumber 2 K: K7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='5 K6 K5 K4 K10 K4 K: K5 K11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='5 K2 Ki 0 K6 K12 100 200 300 400 500 600 Unit Cell No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' EXTENDED RESULTS FOR WAVENUMBER ENTROPY The results of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 5 were recovered for the entire ensemble of simulations conducted for the diatomic (2-band) lattice of section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The entire ensemble considered VI configurations depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 1 for excitation wavenumbers ranging from 2π/9 to 7π/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The resulting normalized wavenumber entropy trends with respect to input forcing are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' S2 for all simulations, where it is seen that the trends presented in section II are agnostic to the excitation wavenumber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Power law fits are superimposed onto each subplot, and the adjusted R-squared values of the fits range between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='9 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='99 for nearly every simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Wavenumber entropy versus excitation amplitude for all datasets generated for the diatomic lattice system of section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Note that the notation ˜u indicates displacement defined over the Bloch-eigenmode, not to be confused with the notation u which FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' the red squares the optimal pairing of the parameters (λ,η), and the insets depict the resulting dispersion relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 4 corresponds to coordinate displacements of the finite lattice in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' (S1) This gives four Bloch-eigenmode solutions for x(κ) corresponding to the four bands of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The resonator of the 4 DoF model is described by ˜u2(κ) and ˜u3(κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Moreover, the cost functions of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' S3 show trivial minimum and maximum solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' ADDITIONAL RESULTS FOR IBTET A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The phase trajectories across regions I-IV of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' To do this, simulations were considered whereby only one VI unit cell is embedded in the lattice with either Hertzian or bilinear contact law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Low energy orbits are smooth and circular, indicating a linear response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 7 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Detailed simulation response for bilinear system Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' For completeness, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' S6 depicts the same computational summary computed for a system with embedded bilinear VI nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The same remarks stated for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 7 in the main text apply to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' S6 as well, further corroborating the similarities in behavior between Hertzian VIs and bilinear VIs with respect to IBTET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' IBTET in the 4-band lattice with bilinear VI nonlinearity and 5 VI sites: (a) shows the evolution of the propagating wave energy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' (b-e) propagation of the wave energy corresponding to each band of the lattice based on the numerically recovered dispersion of the full simulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' (f,g) dispersion of the input and output segments (labeled in (a)) demonstrating the targeted energy transfer to the higher bands;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 8 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The results are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' S7 where very similar trends to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 12 are recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Moreover, the effect of having only a single VI unit cell configuration is was considered as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' To this end, another set of simulations was performed subject to the 30 cycle excitation as the case for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 12 of the main text, but now for only 1 VI embedded within the finite lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The resulting IBTET are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' S8 with the normalized FEP slope superimposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The same trends are recovered again, but with some minor differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The total energy transfer achievable is unsurprisingly less (maxing out at approximate 10 percent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' Hence, the normalization constants for the FEP slopes are slightly different, which is why the FEP slopes superimposed appear slightly different in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' In other words, the energy transfer is dependent on the momentum transfer of incident waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' With additional VIs, this momentum transfer is better averaged out across the system as compared to the single VI case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' However, the agreement in the overall trends of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' S8 supports the arguments developed in section IV for the evolution of the BTET mechanism with respect to system energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 12, but for 15 cycles of input excitation instead of 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The relative energy inter-band energy transfer, with the normalized slope from the ROM-FEP superimposed for (a) Hertzian and (b) bilinear contact models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 12, but for 1 VI engaged instead of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' The relative energy inter-band energy transfer, with the normalized slope from the ROM-FEP superimposed for (a) Hertzian and (b) bilinear contact models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQf3A1I/content/2301.05302v1.pdf'} diff --git a/9dFAT4oBgHgl3EQfpx0O/content/tmp_files/2301.08641v1.pdf.txt b/9dFAT4oBgHgl3EQfpx0O/content/tmp_files/2301.08641v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ce09286c60f941f7662de0b382d69f712d078328 --- /dev/null +++ b/9dFAT4oBgHgl3EQfpx0O/content/tmp_files/2301.08641v1.pdf.txt @@ -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 +qqΦ† +qΦq + m2 +eeΦ† +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. +32 + +[1] G. +Aad +et +al. +[ATLAS +Collaboration], +Phys. +Lett. +B +716 +(2012) +1 +doi:10.1016/j.physletb.2012.08.020 [arXiv:1207.7214 [hep-ex]]. +[2] S. +Chatrchyan +et +al. +[CMS +Collaboration], +Phys. +Lett. +B +716 +(2012) +30 +doi:10.1016/j.physletb.2012.08.021 [arXiv:1207.7235 [hep-ex]]. +[3] G. 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C 45, no.7, 073114 (2021) doi:10.1088/1674- +1137/abf828 [arXiv:2101.06862 [hep-ph]]. +35 + diff --git a/9dFAT4oBgHgl3EQfpx0O/content/tmp_files/load_file.txt b/9dFAT4oBgHgl3EQfpx0O/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a435ab19e3cfee94d31f32dfd03afd0dcf559731 --- /dev/null +++ b/9dFAT4oBgHgl3EQfpx0O/content/tmp_files/load_file.txt @@ -0,0 +1,1196 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf,len=1195 +page_content='Lepton Flavor Specific Extended Higgs Model B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Gon¸calves Departamento de F´ısica and CFTP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Instituto Superior T´ecnico,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Universidade de Lisboa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Lisboa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Portugal and Centro de F´ısica Te´orica e Computacional,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Faculdade de Ciˆencias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Universidade de Lisboa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Campo Grande,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Edif´ıcio C8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 1749-016 Lisboa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Portugal Matthew Knauss and Marc Sher High Energy Theory Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' William & Mary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Williamsburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' VA 23187,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' USA (Dated: January 23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 2023) Abstract In extended Higgs models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' a discrete symmetry is needed in the quark sector to avoid tree-level flavor-changing neutral currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' However, this is not necessary the case in the lepton sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' We consider a model in which one Higgs couples to quarks and three others couple to the electron, muon and tau, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' This four-doublet model is presented with the full scalar potential and the gauge and Yukawa couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' We also consider bounds from searches for heavy neutral and charged scalars at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Since the Standard Model Higgs couplings match predictions very well, we focus on the alignment limit of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='08641v1 [hep-ph] 20 Jan 2023 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' INTRODUCTION The Higgs boson was initially discovered [1, 2] through its decay into gauge bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Since then, the coupling of the Higgs to third generation fermions has also been determined with increasing accuracy [3–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' However, while there is evidence [8] of the Higgs decay into muons, there remain large uncertainties and the discovery has not yet been made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' This leads one to ask if there are viable models in which the muon and tau couple to different Higgs bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' However, the seminal papers of Glashow and Weinberg [9] and of Paschos [10] explicitly referred to the quark sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' As we will see, FCNC can be avoided in the lepton sector even if different leptons couple to different Higgs bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' The first such model, called the muon-specific Two Higgs Doublet (2HDM) model, was developed by Abe, Sato and Yagyu [11] (ASY).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' They use a Z4 symmetry, under which the muon and tau have different quantum numbers, and break this softly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Their model can address the g-2 anomaly, but only for a very 2 narrow region of parameter-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' A more detailed analysis was carried out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' [12] where the phenomenology of the model was studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' All other fields have charge +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' This then has Φ1 coupling to muons and Φ2 coupling to all other fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' This does not affect the ASY Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' One might be concerned about how the PMNS matrix is generated if the muon and muon neutrino mass matrices decouple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' The latter is arbitrary and so the light neutrino mass matrix is not diagonal, leading to a non-trivial PMNS matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Note that this will not work in the quark sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Equivalently, one can replace the Z4 with 3 Z2 × U(1) as discussed above - the Lagrangian in either case is identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' To achieve a non-trivial PMNS matrix, the symmetry must be softly broken in the superheavy Majorana neutrino mass matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' The simplest implementa- tion of this model would be a 4HDM in which the fourth Higgs Φq couples to the quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' This is similar to the lepton-specific model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Certainly one could have one of Φℓ be the same as Φq, leading to a 3HDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' For simplicity, we assume they are separate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' So, we will focus on the 4HDM with Φq, Φe, Φµ and Φτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Although there are hundreds of papers that study models with three Higgs doublets, very few look at models with four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' A supersymmetric model [15] had one doublet each coupling to up-quarks, down-quarks and leptons, with the fourth needed for anomaly cancellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' A similar non-supersymmetric model was proposed [16](with the fourth Higgs needed to relax some tight constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' An early discus- sion that mentions 4HDMs [17] studied Abelian symmetries in multidoublet models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' There are also many studies of symmetries and vacuum states of N doublet models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Most rel- 4 evant papers before that time are referred to in this review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' A more recent paper [19] looked at the interesting issue of non-decoupling in multiscalar models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Related work [20] dealt with large discrete symmetry groups in N doublet models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Additionally, the “Private Higgs” model of Porto and Zee [21, 22] had one Higgs doublet for every fermion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' In contrast to the model we propose, their model had numerous discrete symmetries and included several “darkon” scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' In section II, the model is presented, including the full scalar potential and the gauge and Yukawa couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' In section III, we discuss the constraints on the potential from boundedness and constraints from oblique parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' In section IV, two benchmark models are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Section V contains our results and conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' THE MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Scalar sector The potential can be written as a sum of quadratic and quartic terms: V = V2 + V4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' We allow for soft breaking of the discrete symmetry in the quadratic terms: V2 = m2 qqΦ† qΦq + m2 eeΦ† 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='V4 = λq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='1(Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='qΦq)2 + λe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='1(Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='eΦe)2 + λµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='1(Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='µΦµ)2 + λτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='1(Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='τΦτ)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='+ λqe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='3 (Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='qΦq)(Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='eΦe) + λqµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='3 (Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='qΦq)(Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='µΦµ) + λqτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='3 (Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='qΦq)(Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='τΦτ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='+λeµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='3 (Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='eΦe)(Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='µΦµ) + λeτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='3 (Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='eΦe)(Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='τΦτ) + λµτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='3 (Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='µΦµ)(Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='τΦτ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='+ λqe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='4 (Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='qΦe)(Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='eΦq) + λqµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='4 (Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='qΦµ)(Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='µΦq) + λqτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='4 (Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='qΦτ)(Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='τΦq) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='+ λeµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='4 (Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='eΦµ)(Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='µΦe) + λeτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='4 (Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='eΦτ)(Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='τΦe) + λµτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='4 (Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='µΦτ)(Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='τΦµ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='λqe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='5 (Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='qΦe)2 + λqµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='5 (Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='qΦµ)2 + λqτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='5 (Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='qΦτ)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='+ λeµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='5 (Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='eΦµ)2 + λeτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='5 (Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='eΦτ)2 + λµτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='5 (Φ† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='µΦτ)2 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' � (2) Here, we have labeled the quartic couplings to be similar to the standard 2HDM potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' To discuss diagonalizing mass matrices and the various angles involved, we follow the procedure of Boto, Rom˜ao and Silva [23] closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Without loss of generality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' we can define the angles that rotate the fields ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='into the Higgs basis in which only one scalar field gets a vev by ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='vq = v cos β2 cos β3 cos β4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ve = v sin β2 cos β3 cos β4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='vµ = v sin β3 cos β4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='vτ = v sin β4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='(4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='giving ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='h0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='H1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='H3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='= Oβ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='φq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='φe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='φµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='φτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='(5) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='Oβ = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='cβ2cβ3cβ4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='sβ2cβ3cβ4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='sβ3cβ4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='sβ4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='−sβ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='cβ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='−cβ2cβ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='−sβ2sβ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='cβ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='−cβ2cβ3sβ4 −sβ2cβ3sβ4 −sβ3sβ4 cβ4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='(6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' h0 is the field that gets the entire vev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' and cθ (sθ) are cos θ (sin θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' From this basis, we can now diagonalize the mass matrices of the various scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' We see that there are six rotation angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Gauge and Yukawa couplings 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' The couplings ZZhi and W ±W ∓hi are written in the form � 4 � i=1 Cihi � � g 2cW mZZµZµ + gmWW − µ W +µ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' (15) The Ci factors are included in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Yukawa couplings Following the notation of Branco, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' [24],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' the couplings of the scalar and pseudoscalar Higgs are defined through LS Y = − � f∈{q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='τ} mf v � ξf h1 ¯ffh1 + ξf h2 ¯ffh2 + ξf h3 ¯ffh3 + ξf h4 ¯ffh4 � LP Y = − � f∈{q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='1 ˆv1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' ξe hj = Oαj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='2 ˆv2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' ξµ hj = Oαj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='3 ˆv3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' ξτ hj = Oαj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='4 ˆv4 ξq Aj = (OγOβ)j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='1 ˆv1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' ξe Aj = (OγOβ)j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='2 ˆv2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' ξµ Aj = (OγOβ)j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='3 ˆv3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' ξτ Aj = (OγOβ)j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='4 ˆv4 (17) using ˆvi ≡ vi/v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' THEORETICAL CONSTRAINTS ON THE SCALAR POTENTIAL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' The case in which the potential turns over at very high scales due to renormalization group running will not be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' In fact, the Standard Model itself would not satisfy that latter condition 10 more than two doublets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' [25] for an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' A recent discussion of these conditions for a three-doublet model can be found in the work of Boto, Rom˜ao and Silva [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' We will first discuss the neutral directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' As shown in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' [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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' For the 4HDM that we consider, the corresponding conditions must be satisfied for every three dimensional subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' The remaining conditions are extremely 11 complicated, but are given in full in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' We have incorporated the conditions in that paper to ensure stability in the neutral directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' This requirement is, of course, necessary but may not be sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Oblique Parameters To discuss the S, T, U oblique parameters, we follow the methods and results in Grimus, et al [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' To do this, we can write the matrices ˜U and ˜V from Grimus, et al [31] using our notation in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' ˜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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' (27), our notation slightly differs from Grimus et al [31] by keeping the Goldstone boson with the pseudoscalar mass eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' ˜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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='10 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='12 (31) We will not include the detailed calculation of the unitarity and perturba- tivity bounds, due to the large number of scalar couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Rather, we will simply require that all of the quartic scalar couplings be less than 4π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' BENCHMARK MODELS As is clear from examining the scalar potential and the Appendices, the model contains a large number of free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' To focus on the most important aspects of the model, we will consider two benchmark models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' In the first, we will assume that the (qτ) sector of the Higgs potential decouples from the (µe) sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' In that case, the 4 × 4 scalar mass matrices decouple into two 2 × 2 matrices which can be trivially diagonalized analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' In the second benchmark model, we will take the alignment limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' In this case, it is easy to see from Appendices A and B that the alignment limit corresponds to α1j = βj, as previously stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' The upper 2 × 2 matrix looks very similar to the lepton-specific 2HDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' The only difference involves the coupling to the muon, which is not well-measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' As a result, all Yukawa couplings will be increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' This will affect the decays of the 125 GeV Higgs boson as well as the production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='50 1 5 10 50 cos(β-α) tan(β) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='95 in purple, r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='90 in blue and r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='85 in cyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' precision of current data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 2 We see that if the ratio of (v2 q +v2 τ)1/2 to v is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='85, that the entire parameter space practically disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Thus much of the vev is saturated by vq and vτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 15 g2 HQQ, from Higgs production, now combined with both g2 Hqq and g2 Hll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' In itself, this benchmark model is phenomenologically unacceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Only one can be absorbed by the W and Z gauge bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' The additional massless scalars arise due to an additional accidental SU(2) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Thus, there must be some off-diagonal terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' We can include these terms but assume they are small and do a perturbative expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' For simplicity, let us add a single off-diagonal term, λqµ 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' This will allow for nonzero masses for the lightest charged and pseudoscalar Higgs3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' This term will modify the Yukawa couplings of the Standard Model 125 GeV Higgs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' For the couplings of the quarks, for example, the Yukawa coupling gY ¯qqΦq is √ 2mq/vq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Writing Φq = V11h1 + V12h2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=', where h1 is the 125 GeV Higgs, one sees that the coupling is modified by a factor of v vqV11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' The relevant point here is that V11 is reduced, which counters the effect of the smaller vq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 16 to (vµ/mh3)2) is quite small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' The Aligned Model The full 4HDM has a large number of parameters in the scalar potential: 10 quadratic terms and 22 quartic terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Not surprisingly, many of these parameters will have little effect on phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' From Appendix A, we see that this will occur if α1j = βj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' As we will see, many of these parameters will not be relevant for particular processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' For every possible set of parameters, we require the following: The bounded-from-below conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' The perturbativity condition that the absolute values of λ parameters are less than 4π is maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' The previous condition also applies to Yukawa couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' The values of the S and T parameters are within the range given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Charged Higgs masses must exceed 80 GeV [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Contributions from the charged scalars to the loop-induced Higgs dipho- ton decay h → γγ are compatible with experimental bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' This is achieved by checking the value of the diphoton signal strength µγγ [35, 36] for each set of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Such nonstandard contributions come from charged scalars through one-loop processes [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Contributions to b → sγ [39], again from charged Higgs particles, are acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' In the Type II 2HDM, this gives the strongest constraint on charged Higgs bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' At the LHC, CMS [40] has searched for a heavy neutral Higgs decaying into τ pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' More recently, ATLAS [41] has done a similar analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' We impose these experimental bounds on our parameter-space, which, up to small differences due to form factors, apply to neutral scalars and pseudoscalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Finally, we can consider LHC direct searches for heavy charged Higgs bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Searches fall into two categories - those in which the charged Higgs mass is greater than mt + mb and those in which it is less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' – 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='01 picobarns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' ATLAS [43] has found bounds from Run II on the product of the production rate and the H+ → t¯b branching ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' – If the charged Higgs is lighter, then a major decay mode is into τντ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' In this case the predominant production mode is through t → bH+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Since top production is well understood, searches at ATLAS [44] and CMS [45] place bounds on BR(t → bH+)BR(H+ → τντ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Nonetheless, we have incorporated the results of these searches in bounding our parameter-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' We will primarily focus on the lightest neutral scalar (other than the 125 GeV Higgs), the lightest pseudoscalar and the lightest charged scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' The lepton-specific 2HDM has one scalar coupling to quarks and another to leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' As a result, we will focus on decays involving muons and electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Since the heavier masses aren’t relevant in the analysis, the parameter-space is substantially reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' We consider two mass regions, in which the scalar mass is below and above 350 GeV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' In the latter case, decays to top quarks can be substantial, even if the mixing angles are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' As noted above, given the masses, soft-breaking mass parameters and mix- ing angles, the quartic couplings are determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' We scan the full parameter space and check each of the conditions above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Typically, we find several mil- lion parameter sets that are acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' The results are plotted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Above 350 GeV, the opening of the top decay channel, even if the mixing angle is very small, substantially reduces the leptonic branching ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' It is not surprising that this can occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' This would require extreme fine-tuning, since no symmetry will eliminate mixing in the quartic sector of 21 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 2: These scatterplots show allowed points for h2 decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Results are shown for h2 masses below 350 GeV and above that mass scale (at which point the ¯tt channel opens up).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' The upper figures plot ee and µµ decays and the lower figures plot µµ and ττ decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' The decay branching ratio of the SM Higgs to µµ is approximately 2 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' the potential and even very small values of the quartic mixing terms would allow for other decays that could dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' In Figure 2, we also show the branching ratios to muons and to taus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Again, one can see that the absolute branching ratio to dimuons can be substantially more than that into two taus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 10-2 ee) BR(h2 →ee) 10-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 10-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' BR(h2 →( 10-4 10-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 10-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 10-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 10-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 10-2 10-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' (nn (nn← 个 BR(h2) 10-3 10-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 10-4 10-5, 10-5- 10-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 10-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 3: These scatterplots show allowed points for H± decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Results are shown for H± masses below 180 GeV and above that mass scale (at which point the ¯tb channel opens up).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' The upper figures plot eν and µν decays and the lower figures plot µν and τν decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' tauonic decays, should also study muonic and electronic decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Since we are in the alignment limit, there is no three-point coupling of these scalars to two gauge bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' They could be produced in a collider through WW or ZZ fusion to two Φs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' The signature would be two electron- positron or muon pairs each coming from a Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' The electron-positron pair rate will be smaller, but more distinctive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' While four lepton events have been searched for [46], we know of no analysis of this particular signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' An approximate production cross-section can be obtained by comparison with the inert doublet model[47] which has a similar production process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 23 100 100 mH < 180 GeV mH# > 180 GeV 10-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 10-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 10-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' e= et, 10-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 个 10-3 R 10-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 10- B 10-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 10-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 10-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 10-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 10-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' >10-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 10-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 个 10-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 10-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' +1 BR 5 10-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 10-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 10-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 10-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' mH# < 180 GeV mH± > 180 GeV 10-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 10-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='5 fb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' We have also studied the decays of the pseudoscalar into leptons and find very similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Here, we consider mass ranges below and above 180 GeV, at which point the t¯b opens up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' There is a large number of points in which the electronic decays are substantial and the muonic decay branching ratios can approach unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' In Appendix C we show several benchmark points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' These points satisfy all of the various constraints listed earlier in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' For point S1, one can see that the h2 → µµ branching ratio is almost 47% and the electronic branching ratio is over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='25%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' In benchmark point S2, the dimuon decay of the scalar is smaller than that of the electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Here, one would see the ditau decays dominate, but the electron-positron decays might be measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' For point C1, the decay into muons is slightly bigger than the decay into taus, and the electronic decay is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' For C2, the muon 24 decay is the smallest and the electron decay is as high as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Again, this shows that decays into muons and electrons might be much, much higher than in traditional 2HDMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' However this is only true in the quark sector and need not be true in the lepton sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' The results for the lightest pseudoscalar and charged scalar are also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Generally, searches for heavier Higgs bosons focus (in the lepton sector) on decays into τs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' However, this model shows that decays into electrons and muons can be substantial (and certainly easier to detect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' We know of no bounds on such a process and hope to see searches in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Acknowledgments The work of MS and MK was supported by the National Science Foun- dation under Grant PHY-1819575.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' BLG thanks the Ful- bright Commission in Portugal and William & Mary for support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' MK thanks Pitt-PACC at the University of Pittsburgh for their hospitality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='Appendix A: Gauge Couplings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='Trilinear Gauge Couplings ZZhi and W ±W ∓hi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='C1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='c12c13c14c2c3c4 + c13c14c3c4s12s2 + c14c4s13s3 + s14s4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='−c12c2c3c4(c24s13s23 + c13s14s24) − c23c24c3c4s12−2 − c24c3c4s12s13s23s2 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='c13c3c4s12s14s24s2 + c13c24c4s23s3 − c4s13s14s24s3 + c14s24s4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='C3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='−c12c3c4[c13c24c2s14s34 + s23(−c2s13s24s34 + c34s2) + c23(c34c2s13 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='s24s34s2)] + c34c4[c2c3s12s23 + c23(−c3s12s13s2 + c13s3)] + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='s34[c23c2c3c4s12s24 + c3c4s12s13s23s24s2 − c24c4s13s14s3 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='c13c4(c24c3s12s14s2 + s23s24s3) + c14c24s4] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='C4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='−c2c3c4s12s23s34 − c13c24c34c3c4s12s14s2 + c34c3c4s12s13s23s24s2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='c12c3c4[−c13c24c34c2s14 + c34s24(c2s13s23 − c23s2) + s34(c23c2s13 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='s23s2)] − c24c34c4s13s14s3 − c13c34c4s23s24s3 + c23c4[c34c2c3s12s24 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='s34(c3s12s13s2 − c13s3)] + c14c24c34s4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='TABLE I: Ci-factors of the trilinear gauge couplings ZZhi and W ±W ∓hi as defined in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' (15) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Here cij = cos αij (sij = sin αij) and ci = cos βi (si = sin βi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' In this notation, sij−k stands for sin(αij − βk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='Appendix B: General Yukawa Couplings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='General Yukawa Neutral Scalar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξud ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='c12c13c14 / c2c3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='s12c13c14 / s2c3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='s13c14 / s3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='s14 / s4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξud ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='h2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='− (c23c24s12 + c12 (c24s13s23 + c13s14s24)) / c2c3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='h2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='(c12c23c24 − s12 (c24s13s23 + c13s14s24)) / s2c3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='h2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='(c13c24s23 − s13s14s24) / s3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='h2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='c14s24 / s4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξud ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='h3 (s12 (c34s23 + c23s24s34)−c12 (c13c24s14s34 + s13 (c23c34−s23s24s34))) /c2c3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='h3 −(c12 (c34s23+c23s24s34)+s12 (c13c24s14s34+s13 (c23c34+s23s24s34)))/s2c3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='h3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='(−c24s13s14s34 + c13 (c23c34 − s23s24s34)) / s3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='h3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='c14c24s34 / s4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξud ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='h4 (s12 (c23c34s24 − s23s34)−c12 (c13c24c34s14−s13 (c34s23s24 + c23s34))) /c2c3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='h4 −(c12 (c23c34s24−s23s34)+s12 (c13c24c34s14−s13 (c34s23s24+c23s34)))/s2c3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='h4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='− (c24c34s13s14 + c13 (c34s23s24 + c23s34)) / s3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='h4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='c14c24c34 / s4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='TABLE II: General Yukawa couplings of the scalar Higgs particles to quarks and charged ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='leptons,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' as defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' (16) and (17) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Here cij = cos αij (sij = sin αij) and ci = cos βi (si = sin βi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='General Yukawa Pseudoscalar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='A1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='− (c23c24s2 + c2 (c24s3s23 + c3s4s24)) / c2c3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='A1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='(c2c23c24 − s2 (c24s3s23 + c3s4s24)) / s2c3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='A1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='(c3c24s23 − s3s4s24) / s3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='A1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='s24c4 / s4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='A2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='(s2 (c34s23 + c23s24s34) − c2 (c3c24s4s34 + s3 (c23c34 − s23s24s34)))/c2c3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='A2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='−(c2 (c34s23 + c23s24s34)+s2 (c3c24s4s34 + s3 (c23c34−s23s24s34)))/s2c3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='A2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='(−c24s3s4s34 + c3 (c23c34 − s23s24s34)) / s3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='A2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='c24s34c4 / s4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='A3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='(s2 (c23c34s24 − s23s34) − c2 (c3c24c34s4 − s3 (c34s23s24 + c23s34)))/c2c3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='A3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='−(c2 (c23c34s24 − s23s34)+s2 (c3c24c34s4−s3 (c34s23s24 + c23s34)))/s2c3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='A3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='− (c24c34s3s4 + c3 (c34s23s24 + c23s34)) / s3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='A3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='c24c34c4 / s4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='TABLE III: General Yukawa couplings of the pseudoscalar Higgs particles to quarks and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='charged leptons,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' as defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' (16) and (17) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Here cij = cos γij (sij = sin γij) and ci = cos βi (si = sin βi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='General Yukawa Charged ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξqLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='H+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='− (c23c24s2 + c2 (c24s3s23 + c3s4s24)) / c2c3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξeL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='H+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='(c2c23c24 − s2 (c24s3s23 + c3s4s24)) / s2c3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξµL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='H+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='(c3c24s23 − s3s4s24) / s3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξτL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='H+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='s24c4 / s4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξqLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='H+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='(s2 (c34s23 + c23s24s34) − c2 (c3c24s4s34 + s3 (c23c34 − s23s24s34)))/c2c3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξeL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='H+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='−(c2(c34s23 + c23s24s34)+s2 (c3c24s4s34 + s3 (c23c34 − s23s24s34)))/s2c3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξµL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='H+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='(−c24s3s4s34 + c3 (c23c34 − s23s24s34)) / s3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξτL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='H+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='c24s34c4 / s4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξqLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='H+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='(s2 (c23c34s24 − s23s34) − c2 (c3c24c34s4 − s3 (c34s23s24 + c23s34)))/c2c3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξeL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='H+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='(c2 (s23s34 − c23c34s24) − s2 (c3c24c34s4 − s3 (c34s23s24 + c23s34)))/s2c3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξµL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='H+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='− (c24c34s3s4 + c3 (c34s23s24 + c23s34)) / s3c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='ξτL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='H+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='c24c34c4 / s4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='TABLE IV: General Yukawa couplings of the charged Higgs particles to quarks and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='leptons,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' as defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' (18) and (19) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Here cij = cos δij (sij = sin δij) and ci = cos βi (si = sin βi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 30 Appendix C: Benchmark Points Scalar benchmark points S1 S2 β2/π, β3/π, β4/π 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='16, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='14, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='21 α23/π, α24/π, α34/π −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='09, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='00, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='70 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='02, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='10 γ23/π, γ24/π, γ34/π 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='50, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='59, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='16, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='52, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='39 δ23/π, δ24/π, δ34/π 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='08, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='26, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='62, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='93, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='72 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='63 × 10−4 BR(h2 → µµ) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='68 × 10−1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='85 × 10−6 BR(h2 → ττ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='22 × 10−1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 31 Charged benchmark points C1 C2 β2/π, β3/π, β4/π 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='16, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='11 α23/π, α24/π, α34/π 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='09, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='54, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='20, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='88, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='72 γ23/π, γ24/π, γ34/π −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='66, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='68, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='50, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='52 δ23/π, δ24/π, δ34/π −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='98, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='00, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='36 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='00, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='00, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='24 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='68 × 10−2 BR(H± 1 → µ±νµ) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='36 × 10−1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='91 × 10−3 BR(H± 1 → τ ±ντ) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='55 × 10−1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 32 [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Aad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' [ATLAS Collaboration], Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' B 716 (2012) 1 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='physletb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='08.' 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1137/abf828 [arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content='06862 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} +page_content=' 35' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFAT4oBgHgl3EQfpx0O/content/2301.08641v1.pdf'} diff --git a/AtAyT4oBgHgl3EQfq_mZ/content/tmp_files/2301.00553v1.pdf.txt b/AtAyT4oBgHgl3EQfq_mZ/content/tmp_files/2301.00553v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..635ec9c6a8b59903fc8aa3359f6df9394971a78b --- /dev/null +++ b/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. +6. REFERENCES +[1] K. Nazeri, E. Ng, T. Joseph, F. Z. 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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 \ No newline at end of file diff --git a/AtAyT4oBgHgl3EQfq_mZ/content/tmp_files/load_file.txt b/AtAyT4oBgHgl3EQfq_mZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4024667490ddb70a86a6fe966e699fd517417e42 --- /dev/null +++ b/AtAyT4oBgHgl3EQfq_mZ/content/tmp_files/load_file.txt @@ -0,0 +1,780 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf,len=779 +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' As admirable methods are presented, the inpainted image is get- ting closer to reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' However, the result is still not good enough in the reconstructed texture and structure based on human vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Therefore, we propose a lightweight model that combines the special transformer and the traditional convolutional neural network (CNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Index Terms— HSV color space, image inpainting, joint attention mechanism, stripe window, vision transformer 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' INTRODUCTION Image inpainting has been studied by many researchers for several years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' To achieve realistic results, we need to consider the follow- ing two important points: 1) the continuity of adjacent tex- tures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 2) visually reasonable structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Without the global infor- mation of the whole image, it is hard to repair the key edge and lines of the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Some researchers proposed methods that utilize auxiliary information for structure recovery, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=', edges [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Suvorov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' [2] utilize the Fast Fourier Convolution (FFC) to encode features in the frequency domain with global receptive fields for resolution-robust inpainting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Although these methods im- prove the overall repair result but also causes a huge compu- tational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Furthermore, in recent years, the transformer has also been used in the inpainting field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' It has the advantage of wider receptive fields than CNNs and better inpainting at low resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Unfortunately, transformers require a lot of computer memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Therefore, it inspires us to design a lightweight trans- former block with stable repair effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Specifically, we use the CSWin transformer [3] which used stripe window self- attention to replace the traditional full self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Stripe window self-attention mechanism computes self-attention parallel to horizontal and vertical stripe cross-windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Each stripe is obtained by dividing the input feature into constant- width stripes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' In this way, we can achieve global attention with limited computational cost and we redesign the trans- former block to improve the repair performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' The consistency of color is another important factor to judge the quality of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' It is easy to discern the differ- ence between inpainted image and original image by human vision if the color has deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Most researchers only deal with basic primary colors but we think this is not enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' If we can quickly improve color consistency in the early stage of training, the repair performance can be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Therefore, we transform the inpainted image to HSV color space and compare it with the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' In follow-up experiments, our method is confirmed to be effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' In Section 2, we introduce the previous and state-of-the-art inpainting methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Then we present our proposed method and loss function in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' In Section 4, we exhibit our training details, experiment results, inpainting images, and ablation studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' At last, the conclusions are drawn in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='00553v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='IV] 2 Jan 2023 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' RELATED WORK Traditional inpainting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Traditional inpainting can gener- ally be divided into two categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' The first one is the dif- fusion method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Diffusion methods disseminate the texture content by one or multi-curve information from the known region to missing region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' The second one is the patch- matching method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' The former method is easy to blur inpainting results in large masks, and the latter method will cost a lot of calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Deep learning based inpainting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' With the advancement of hardware technology, CNN based deep learning model has become the mainstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Gradually, more and more novel CNN models based on different modules have been proposed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=', some models utilize edge auxiliaries information, such as Nazeri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' proposed Edgeconnect [1], Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' proposed GateConv [5] which used Canny edge to generate edge im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' We also used edge information for the mask instead of the input in our proposed approach to enhance the edge structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' They calculate complicated attention scores to find the most similar texture that can be filled in the missing region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Gen- erally speaking, this type of methods is better than others in terms of texture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' In our proposed method, we redesign the attention module and combine wide attention to the local re- ceptive field to achieve attention sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Vision transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Recently He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' proposed a model named Vision transformer (VIT) [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Transformer has been long and widely used in the field of NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' They made the trans- former usable in computer vision by their proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' As more novel transformers are proposed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=', 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' For huge mask, transformer can inpaint plausible textures by their special attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' In addition, transformer has wider re- ceptive field than traditional convolution but also needs more computing costs than convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Our model focuses on the global CSWin transformer and CNN-based local layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' We process the global and local layer in parallel and then share the same attention information be- tween them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' In the end, we use four simple up-samples to get the inpainting result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' The major contributions of this work are as follows: We propose a stripe window self-attention transformer with an efficient local enhancement position encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Then we redesign the transformer block to make the result better than the original method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' We suggest joint attention from global layers to local layers, connecting the two layers to enhance the overall consistency of repair results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' We propose a new HSV loss focused on color consis- tency in the early stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' In the common dataset including Places2, we conduct extensive experiments to confirm that our proposed model is better than other advanced methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' METHODOLOGY Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' The whole model of our proposed approach is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' After we downsam- ple input image, we split the channel to global layer (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=', 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Each RDB block in RRDB has four consecutive Conv-ReLU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' At last we concate- nate the features from both channels and then go through three upsample layers to get the inpainted image Iout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Special CSWin Transformer The overall global layer special CSWin Transformer is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' There are four CSWin transformer blocks in our global layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Each block has its own multi-head and stripe window (sw) to reduce the amount of calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' We set multi-head to 2, 4, 8, 16 and sw to 4, 8, 16, 32 for four blocks by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' The first three blocks are our special CSWin transformer block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' They will split their channel into horizontal and vertical stripes, and then split their channel with their own multi-head again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' The sw will split H or W chosen by horizontal stripes or vertical stripes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' After we get the split low-resolution image, we can do the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 1: The overview of our proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' The whole model structure shows the framework of our proposed model and the details of the joint attention between Global layer and Local layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Moreover, the right side shows the CSWin Transformer Block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' D is the normalization factor before softmax, which makes the similarity between pixels become more stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' At last, the Residual Dense Block in the local layer is shown at the top right corner of the whole model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' self-attention through Q(query), K(keys), V (values) until the last block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Redesigned CSWin Block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' The structure of CSWin Block is also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Stripe Window Self-Attention and Full Self-Attention will be trained from different receptive fields and then connected together with the residual link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' We found that self-attention needs to be calculated multiple times to get better attention information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' We set the Ni to denote the number of repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Joint attention We concatenate global and local layers to jointly focus on the information with different receptive fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' We expect our inpainting results to be the admixture of different recep- tive fields, not only just global but also local receptive fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' So we collect attention from the second and fourth CSWin transformer blocks and multiply it by the corresponding RDB blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' At last two mixed receptive fields are added to the respective last block of the two layers to achieve joint attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Loss Function Most loss functions we adopt are the same as [1,13,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' And we also use other losses including Edge loss and HSV loss which we proposed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' First, the basic L1 func- tion is described as L1 = |Iout − IGT |, where Iout, IGT indicate predicted images and the ground truth, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' The Iedge is the image obtained from Canny edge detection [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' In order to improve the quality of the inpainting model, we use Perceptual loss to measure the similarity between images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' We also use the mask on feature map to let our Perceptual loss only focus on visible regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' According to [14], checkerboard artifacts are usually caused by deconvolution and using Style loss can remove this artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Therefore, we use the same Style loss as [14] in our total loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Besides focusing on texture and structure, we believe that color is as important as both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' So we proposed the HSV loss to measure the similarity between colors, which can be formu- CSWin Transformer MLP2 CNN CNN >SoftMax( - 七.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' LHSV edge = 1 n n � i=1 ||(HSVout ⊙ Medge − HSVGT ⊙ Medge)||2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' LT otalHSV =λHSV ∗ LHSV + λHSV edge ∗ LHSV edge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' (1) where λHSV = 10 and λHSV edge = 100 by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' If we still use the V alue in HSV loss it will even affect our inpainting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' We demonstrated this in ablation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' The adversarial loss includes the discriminator loss LD and the generator loss LG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' The LGP = EIGT || ▽IGT D(IGT )||2 is the gradient penalty and λGP = 1e − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='1, λstyle = 250, λT otalHSV = 1, and λadv = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' The above loss weights are empirically set by experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' EXPERIMENTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Datasets To show the inpainting effectiveness of our proposed model, we conduct experiments on Places2 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' For all of the images in Places2 dataset, we only train and test them with image size 256×256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' For other comparison methods, we use their provided pretrained model to perform the test on the same dataset as we did.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Reference State-of-the-Art We compare the proposed model with other state-of-the-art methods,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' which include PatchMatch (PM) [4],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Contextual At- tention (CA) [6],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Shift-net (SN) [16],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Partial Convolutions (PC) [14],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Region-wise (RW) [17],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Gated Convolution (Deep- Fill v2) [5],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Contextual Residual Aggregation (HiFill) [8],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Im- puted Convolution (Iconv) [18],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Aggregated contextual trans- formations (AOT-GAN) [19],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Mask-Aware Dynamic Filter- ing (MADF) [7],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Auxiliary Contextual Reconstruction (CR- Fill) [20],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Bridging Global Context Interactions (TFill) [10] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Large Mask inpainting (LaMa) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' For Places2, our proposed method can defeat most of com- pared methods in terms of these two evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' On the other hand, our training images and steps are also less than most methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Hence, our proposed model will surpass them if we have similar resources as they do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' The perceptual similarity is more like what human vision sees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' For Places2, our proposed model can achieve the best results among all compared methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' This means our inpainting images are closer to the real than other compared methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Qualitative Comparisons We show the qualitative inpainting results of Places2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Compared with other methods, our proposed model can reconstruct similar or even more clear textures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' We notice our inpainting results are slightly blurred when we focus more on the transformer and less on CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' In the future, we will set restrictions on the local layers so that local information will not be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Note that both our training data and steps are less than other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' We test the stability of the CSWin transformer and the redesign in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' We report Peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' The ▲ denotes larger, and ▼ denotes lesser of the parameters compared to our proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' (Bold means the 1st best;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Underline means the 2nd best;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='8734 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='5227 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='7799 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='2039 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='3965 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='9213 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='7343 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 2: Qualitative results of Places2 dataset among all compared models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' From left to right: Masked image, RW [17], DeepFill v2 [5], HiFill [8], Iconv [18], AOT-GAN [19], CRFill [20], TFill [10], and Ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Zoom-in for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='149 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='155 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='161 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='149 MADF [2021] CRFill [2021] TFill [2022] LaMa [2022] Ours [2023] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='139 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' We noticed the Value (V) of HSV can easily be learned in L1 and other losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' If we still consider V in LHSV , it will in- fluence the balance of the inpainting result, as shown in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' We show the color deviation between with and with- out LT otalHSV in early training steps in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='7935 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='8072 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='1242 w/o redesigned CSWin 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='1027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='8377 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='1221 ours w/o HSV loss 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='2786 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='8459 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='1212 ours w/ full HSV loss 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='4757 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='8541 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='1184 ours w/ redesigned CSWin and HSV loss (w/o V) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='5801 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='8611 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='1156 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 3: Ablation study of color deviation on inpainted images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' From left to right: Masked images, w/o TotalHSV loss, and TotalHSV loss (w/o V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' LT otalHSV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' CONCLUSION In this paper, we propose a lightweight joint attention trans- former architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' We use transformer-based architecture to get wide receptive field information and cooperate with local layers with RRDB by joint attention with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Our pro- posed HSV loss can stabilize the colors in early training steps and eventually further improve the inpainting performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' We use the CSWin transformer and redesign the transformer block to not confuse the two self-attentions and achieve sig- nificant improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Those large models do have an ad- vantage in details but not every researcher has enough hard- ware support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Therefore we propose this approach to demon- strate small models are also able to compete with large mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' REFERENCES [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Nazeri, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Ng, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Joseph, F.' metadata={'source': 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Shechtman, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Wang, “The unreasonable effectiveness of deep features as a perceptual metric,” in Proceedings of the IEEE conference on computer vision and pattern recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 586–595, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' APPENDIX 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' For all of the images in CelebA dataset, we only train and test them with image size 256×256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' For other comparison methods, we use their provided pretrained model to perform the test on the same dataset as we did.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Result We show the qualitative comparison in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' The required param- eters are also shown below each method, where the results are tested by ourselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Although our proposed method lose slightly in tiny masks (5% to 10%), we can defeat most of compared methods in huge masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' On the other hand, our training steps are also less than most methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Hence, our proposed model will surpass them if we have similar re- sources as they do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' We show the qualitative inpainting re- sults of CelebA in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' For CelebA, our inpainted results are slightly different from the ground-truth image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' This hap- pens with too much focusing on global information and no limitation on local information filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Other inpainting images We also exhibit more inpainting results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 6 (Places2) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 7 (CelebA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' From top to bottom is small masks to the huge masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Zoom-in for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Object Removal We additionally conduct object removal experiments in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Our proposed method did well in target removal and background repair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' If the background is relatively single, the result will be better than the grid background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' This means our model needs to enhance structure in inpainting images, which will be our future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Our codes are released in https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='com/bobo0303/LIGHTWEIGHT-IMAGE- INPAINTING-BY-STRIPE-WINDOW-TRANSFORMER- WITH-JOINT-ATTENTION-TO-CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 4: Object removal (size 256×256) results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' From left to right: Original image, mask, object removal result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 3Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 5: Inpainting (size 256×256) results of all compared models in the CelebA dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' From left to right: Masked image, RW, DeepFill v2, Iconv, AOT-GAN, CRFill, TFill, and Ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Zoom-in for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Table 4: Quantitative evaluation of inpainting on CelebA dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' We report Peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' The ▲ denotes larger, and ▼ denotes lesser of the parameters compared to our proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' (Bold means the 1st best;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Underline means the 2nd best;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='4397 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='4637 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='5820 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='3917 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='5311 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='1646 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='8222 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content='7816 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 6: Other inpainting (size 256×256) results in the Places2 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' From left to right: Masked image, RW, DeepFill v2, HiFill, Iconv, AOT-GAN, CRFill, TFill, and Ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' From top to bottom is 5% mask to 60% mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Zoom-in for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' 7: Other inpainting (size 256×256) results in the CelebA dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' From left to right: Masked image, RW, DeepFill v2, Iconv, AOT-GAN, CRFill, TFill, and Ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' From top to bottom is 5% mask to 60% mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' Zoom-in for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} +page_content=' nupre 50700光' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf'} diff --git a/BNE1T4oBgHgl3EQf9Qa6/content/tmp_files/2301.03555v1.pdf.txt b/BNE1T4oBgHgl3EQf9Qa6/content/tmp_files/2301.03555v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bdf84f3f4816086f59861b7aecc03eb06e41d7d8 --- /dev/null +++ b/BNE1T4oBgHgl3EQf9Qa6/content/tmp_files/2301.03555v1.pdf.txt @@ -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: hans@math.unc.edu +(D. Pezzi) Department of Mathematics, Johns Hopkins. +Email address: dpezzi1@jhmi.edu + +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 \ No newline at end of file diff --git a/BNE1T4oBgHgl3EQf9Qa6/content/tmp_files/load_file.txt b/BNE1T4oBgHgl3EQf9Qa6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4435c7232d640cdddd2c537ae41398c678a7d640 --- /dev/null +++ b/BNE1T4oBgHgl3EQf9Qa6/content/tmp_files/load_file.txt @@ -0,0 +1,694 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf,len=693 +page_content='ENERGY DISTRIBUTION FOR DIRICHLET EIGENFUNCTIONS ON RIGHT TRIANGLES HANS CHRISTIANSON AND DANIEL PEZZI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' In this paper, we continue the study of eigenfunctions on trian- gles initiated by the first author in [Chr17] and [Chr19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' The Neumann data of Dirichlet eigenfunctions on triangles enjoys an equidistribution law, being equidistributed on each side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' The proof of this result is remarkably simple, using only the radial vector field and a Rellich type integrations by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' The equidistribution law, including on higher dimensional simplices, agrees with what Quantum Ergodic Restriction would predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' In the present note, we first show that an “obvious” conjecture fails even for the simplest right isosceles triangle using only Fourier series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Finally we present some numerical computations suggesting the behaviour of eigenfunctions on almost isosceles triangles is quite complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Since waves propagate along straight lines, it is reasonable to expect eigenfunctions to live along straight lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Since waves meeting a smooth boundary transversally reflect according to Snell’s law, eigenfunctions have incoming and outgoing com- ponents as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' And since waves are very complicated near corners and other discontinuities, so are eigenfunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' In this note, we study the distribution of interior energy of eigenfunctions on right triangles, which is a measure of phase space concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' There are several results in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' First, we use integrations by parts to connect interior energy to certain weighted boundary integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' A comparison with 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='03555v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='AP] 9 Jan 2023 2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' CHRISTIANSON AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' PEZZI the results in [Chr17] suggests the eigenfunctions have nice phase space distribu- tion properties, however this appears to be false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' In fact, even on the right isosceles triangle, the weighted boundary integrals are subtle, even though the phase space distribution follows from symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Theoretical results on right triangles Our first set of results is on the right isosceles triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Let T be the right isosceles triangle in the xy-plane, oriented as T = {0 ≤ x ≤ 1, 0 ≤ y ≤ 1 − x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Then there is an orthonormal basis for L2 of eigenfunctions satisfying ∥h∂xu∥2 L2(T ) = ∥h∂yu∥2 L2(T ) = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Our second main result is about phase space distribution on rational right trian- gles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' The result is heavily dependent on the choice of orientation for the triangle, and rotating the triangle changes the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' This result is then meant merely as an example of what one can prove with elementary techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Let Ω ⊂ R2 be a right triangle, and consider the semiclassical Laplace eigenfunc- tion problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Let F1, F2, F3 be the sides as in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' We further assume that Ω is rational, meaning that all angles are rational mul- tiples of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Setup for right triangles Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Suppose Ω is the right triangle oriented as in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Let {uj} be the sequence of orthonormal Dirichlet eigenfunctions on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' There exists a density one subsequence {ujk} such that lim sup k→∞ � Ω |h∂xujk|2dV ≤ 7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' We again emphasize that this estimate is highly dependent on the orientation of the triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Expanding these quantities does not give us much information unless one of α or β is close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' The main idea of the proof is to estimate the mass of h∂xu in strips to that of u in strips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' We then use this and the results from [Chr17] on Neumann data on a whole side to get weak estimates on partial Neumann data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Quantum Ergodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' That is, this subsequence of eigenfunctions distributes evenly both on the domain and in frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' The work of Lindenstrauss [Lin06] shows that quantum ergodicity can hold for the whole sequence of eigenfunctions, called quantum unique ergodicity (QUE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' The work of Hassell [Has10] shows that QUE can fail, so the question of QUE versus non-QUE is very subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 4 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' CHRISTIANSON AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Work of Toth-Zeldtich [TZ12,TZ13] extend these results to interior hypersurfaces, again along a density one subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Let T be a planar triangle with sides A, B, C of length a, b, c respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Consider the (semi-classical) Dirichlet eigenfunction problem (1) and assume the eigenfunctions are normalized (||u||2 L2(T ) = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' However, we stress that the integrals need to be over the whole side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' There are several natural questions that arise based on this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' What can be said about the Neumann data on subsets of sides?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' What about volume integrals over the same domain?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Immediate Questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Based on this result, this paper is concerned with two immediate questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' ENERGY DISTRIBUTION 5 This is just an extension of the equidistribution result to arbitrary subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' A second obvious question would be the following: Question 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Is it true that (6) ∀h > 0, � T |h∂yu|2dV = � T |h∂xu|2dV = 1 2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Connecting boundary integrals to interior energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Let us continue to work with the right triangle given by Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' We duplicate the argument from [Chr17] but with the vector field X = x∂x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' The point is that X = 0 on {x = 0} and X is tangential on {y = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' The normal derivative is then ∂ν = γ−1(∂x + a∂y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Since u = 0 along F0, we have ∂τu = γ−1(a∂x − ∂y)u = 0, or ∂yu = a∂xu on F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Hence ∂νu = γ−1(∂x + a∂y)u = γ−1(1 + a2)∂xu = γ∂xu, so that ∂xu = γ−1∂νu along F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' From [Chr17] we know the integral on the right is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Rearranging, this computation would tell us that � Ω |h∂xu|2dV = 1/2, however this “obvious” conjecture appears to be false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 6 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' CHRISTIANSON AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' PEZZI 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Analytical Results for Right Isosceles Triangle 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Introducing the eigenfunctions on the Right Isosceles Triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' For this section, T will be a triangle in Euclidean space with vertices (0, 0), (1, 0), and (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' For the rest of this paper, we will deal with triangles with vertices at the origin and at (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' We will identify triangles by the x coordinate of the third vertex, which will always be on the positive x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Let T be as previously described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' (11) umn = cmn sin(nπx) sin(mπy) + dmn sin(mπx) sin(nπy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Additionally, by normalization, c2 mn = d2 mn = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' We will show that these functions are exhaustive, satisfy the boundary con- ditions, and satisfy the eigenfunction equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' We achieve this expression by noticing that reflecting T across the line y = 1 − x gives a square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' The eigenfunc- tions of the Laplacian on a square are well known, so we know immediately that this list is exhaustive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' We then just have to check all of the usual requirements to verify these are indeed eigenfunctions on the isosceles triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Clearly x = 0 =⇒ umn = 0 and y = 0 =⇒ umn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' A simple computation gives the eigen- value as h−2 = π2(n2 + m2) which is the same as in the square case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Calculating the Volume Integral for the Right Isosceles Triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' One of the metrics we are interested in is � T |h∂yumn|2dV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' We will refer to this as the “y volume integral” for expository convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' The derivative integrals and the function integrals are related by the equation � T |h∂xumn|2 + |h∂yumn|2dV = � T |umn|2dV = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' As the volume metrics are completely understood in this case, it is natural to investigate the analogous metrics on the boundary as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Later on, the y volume integral will be an important metric throughout this paper as a way to test quantum ergodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Showing Equidistribution fails for subsets of the boundary of the Right Isosceles Triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' As the sum of the data on both halves of the bottom side is constant, we only consider the bottom left face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' This calculation is the same for the left face, as we have xy symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' We always have Il(m, n) + Ir(m, n) = 1 by the result for Neumann data on the entire side in [Chr17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Il = Ir = 1 2 represents equidistribution on the two halves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' There exists m, n such that Il(m, n) ̸= 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Moreover, the subsequence Il(k, k + 1) converges to a value other than 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' On the bottom side the normal derivative is just −∂y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' CHRISTIANSON AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Note that if m + n is even, then 2Il(m, n) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' We will numerically show what all of these values are later on, but to construct our subsequence consider mk = k and nk = k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' This is a subsequence for which the terms multiplied by 1 n−m will have the largest magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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 ≈ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='8183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' It is then immediately the case that Ir(k, k + 1) →≈ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='1817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' This is our subsequence that does not equidistribute on subsets in the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' □ The lack of equidistribution on subsets of the sides, even in the limit, is more sur- prising than the y volume integral result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' This contradicts the original conjecture that there was a uniform distribution in the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' In this case the long term behav- ior of these proportions can be completely described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' The following computation is identical to the previous one but done in generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Let m and n be integers such that n − m = j where j is an odd integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Then we have an explicit formula for Il(m, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' ENERGY DISTRIBUTION 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' We proceed in the same manner as the previous proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' □ This computation also shows that, in the limit, the running average of these two values will both be 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' The subsequence of m, n such that they are separated by a fixed integer is density 0 in the sequence of m, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' This ensures via a straightforward limit argument that the running average of each piece also goes to 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' However, encountering m and n pairs with that separation becomes less and less likely as m and n increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Numerically we have verified all of this with our solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' These two plots are not exactly the same as the direct calculation orders points differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Moreover, the accuracy of the boundary integrals, especially because we are dividing them, is not enough to perfectly align these graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' This result establishes that equidistribution fails even on simple subsets of simple triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' In this next section we will expand this result to state these proportions need not even be bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 10 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' CHRISTIANSON AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' PEZZI (a) Computed Bottom Left Neumann Data for the Right Isosceles (b) Plot of Bottom Left Neumann Data using Derived Formula Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Bottom Bottom Left Neumann Data Plots: Computed and Analytical 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Proof of Theorem 3 In this section, we use the result of Marklof-Rudnick [MR12] to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' We drop the subscript and subsequence notation and simply write u for our density one subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Here γ = (1 + a2) 1 2 is the normalizing constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' That means that on F1, ∂yu = 0, on F2 ∂xu = 0, and on F3, ∂xu = 1 γ ∂νu, ∂yu = a γ ∂ν as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Fix 0 < β < a and δ > 0 independent of h, with δ sufficiently small that 0 < β − δ2 < β + δ < β + δ + δ2 < a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=" Let χ(x) be a smooth function satisfying 306090Triangle-BottomLeftNeumannData-200nodes 6'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='8 0 Data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='7 ofNeumann[ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='6 Fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content="1 0 0 200 400 600 800 1000 1200 EigenvalueNumberDirectCalculationofBottomLeftData 6'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='8 1 of Neumann Data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='4 000000000000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='1 0 0 200 400 600 800 1000 1200 EigenvalueNumberENERGY DISTRIBUTION 11 0 β − δ2 β β + δ β + δ + δ2 a Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' The function χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' See Figure 3 for a sketch of such a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Let Ωβ = Ω ∩ {β − δ2 ≤ x ≤ β + δ + δ2} so that supp χ′ ⊂ Ωβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Further let ˜Ωβ = Ω ∩ {β ≤ x ≤ β + δ} so that χ′ = δ−1 + O(δ) on ˜Ωβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' We have Area(Ωβ) = � 1 − (β−δ2) a + 1 − (β+δ+δ2) a 2 � (δ + 2δ2) = (1 − β a )δ + O(δ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Hence Area(Ωβ) Area(Ω) = (1 − β a)δ a/2 + O(δ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' CHRISTIANSON AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' PEZZI so that (17) � ([−h2∆ − 1, X]u)¯udV ≤ 4(1 − β a) a + O(δ) + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' On the other hand, � ([−h2∆ − 1, X]u)¯udV = � ∂Ω χ(x)(h∂xu)h∂ν ¯udS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' On F1, χ(0) = 0 and on F2, ∂x is tangential, so ∂xu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' On F3, ∂xu = γ−1∂νu, so that � ([−h2∆ − 1, X]u)¯udV = γ−1 � F3 χ(x)|h∂νu|2dS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Putting this together, (18) γ−1 � F3 χ(x)|h∂νu|2dS ≤ 4 a(1 − β a ) + O(δ) + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' We will use (18) to estimate the Neumann data on part of F3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' To that end, let X = (1 − x/a)∂x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' On the other hand, � Ω ([−h2∆ − 1, X]u)¯udV = � ∂Ω (1 − x/a)h∂xuh∂ν ¯udS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' On F1, x = 0 so X = ∂x = −∂ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' On F2, ∂x is tangential, so that Xu = 0 on F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' On F3, we have ∂xu = γ−1∂νu as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' That means � ∂Ω (1 − x/a)h∂xuh∂ν ¯udS = − � F1 |h∂νu|2dS + γ−1 � F3 (1 − x/a)|h∂νu|2dS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Rearranging, we have (20) 2 a � Ω |h∂xu|2dV = 2 a − γ−1 � F3 (1 − x/a)|h∂νu|2dS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' CHRISTIANSON AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' □ Remark 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Optimizing again in the variable (1−β/a) yields (1−β/a) = 1/2, for a bound of 3/4+O(δ)+o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' So even if we knew more aboud energy distribution compared to distribution, the techniques of proof in this paper give an unsatisfactory answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Remark 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Note this is particular to triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Let U ⊂ Ω be an open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Suppose we are interested in {n ≥ Mm} for large M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Then #{m2 + n2 ≤ R2 : n ≥ Mm} ∼ R2/M, so has density ∼ 1/M > 0, but � Ω |h∂xu|2dV ≤ M −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' ENERGY DISTRIBUTION 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' We will investigate the ’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='99 triangle’, or, the triangle with vertices {(0, 0), (0, 1), (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='99, 0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Analytical solutions cannot be found, but we can use numerical techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' The y volume integral plot is no longer constant, and in fact has some noticeable structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' There are at least two, and possibly a third, branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' These branches correspond to subsequences of eigenfunctions whose y volume integrals seem to not approach 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' There is also a large band with sizable separation from 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' These behaviors are discussed numerically in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' The second plot shows the values of Il(m, n) for the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='99 triangle, with the adjust- ment of the bounds of integration from (0, 1/2) to (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='495).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Some of the structure 99Triangle-YVolumeIntegral-425nodes 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='8 Calcuated Y Volume Integral 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='1 0 0 200 400 600 800 1000 1200 EigenvalueNumber99Triangle-BottomLeftNeumannData-425nodes 0 00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='9 ： 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='8 Data 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='7 O 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='1 0 0 200 400 600 800 1000 1200 EigenvalueNumber16 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' CHRISTIANSON AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' PEZZI of the plots is carried over from the y volume integral case, but it is less coherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Similar pictures for other triangles mentioned throughout this paper are in an appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Introducing Running Averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Statements about quantum ergodicity al- low for exceptional zero density subsequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Or mathematically, for density 1 subsequence uik: (23) aj = 1 j j � k=1 � T |h∂yuik|2dV → 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' We can also say something similar about the proportion of Neumann data on a given side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Running Averages of Computed Runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' While numerics are never going to be able to answer questions like this definitively, they can more accurately set expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Here are the average values of different metrics from every run done throughout this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' The ’y volume integral’ and ’Proportion Bottom Left’ metrics are the ones used throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' A larger node count represents an increase in accuracy, but we found diminishing returns in increasing node counts in our numerics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' As such, we considered 200 to be sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Values were computed for the first 1250 eigenfunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' ENERGY DISTRIBUTION 17 Triangle Nodes y volume integral Proportion of Bottom Left .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='99 425 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='4998 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='5083 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='98 200 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='4996 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='5098 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='97 200 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='4996 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='5103 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='96 200 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='4993 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='5097 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='95 200 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='4992 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='5100 The overall trend is consistent across metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' The further we get from the right isosceles triangle, the farther the metrics get from the values quantum ergodicity would predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Percentage of Eigenfunctions Approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' The issue with the methods previously described in this chapter is that they do not get to the heart of what we want.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Statements about the density of sequences are extensions of the familiar discrete concept of percentages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' They are statements about how common we would expect that particular subsequence to be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' A density 1 subsequence, in the high-frequency limit, would be expected to appear for almost every value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' As these are limits, there is substantial wiggle room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' We can use this concept to develop metrics that could indicate whether posi- tive density subsequences of the desired properties exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Suppose we thought the running average of the y volume integrals for the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='99 triangle converged to a value less than .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='5 − ϵ is larger than δ for every N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' If this condition was met, than the subsequence of all eigenfunctions whose y volume integral is less than .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='5 − ϵ would have a density greater than δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' This would show that the domain itself was not ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Of course, there is nothing special about viewing the percentage of eigenfunctions below a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Because we only need a subsequence of positive density, we can consider all eigenfunctions that have y volume integrals sufficiently far away from .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='5 ��� > ϵ for varying tolerances ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' The values for a selection of runs are in the following table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 18 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' CHRISTIANSON AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' PEZZI Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Running Percentage Graph - Shows monotonic and asymptotic behavior Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Running Percentage Graph - Shows monotonic and asymptotic behavior Triangle ϵ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='01 ϵ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='005 ϵ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='001 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='99 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='32 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='64 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='98 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='9 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='0 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='97 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='1 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='7 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='96 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='0 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='6 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='04 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='95 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='7 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='6 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='52 Perhaps more interesting than the exact numerical values are the trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' All of the graphs for all three thresholds for the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='99, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='98, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='97, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='96 and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='95 triangles have the same fundamental shape: increasing with a vertical asymptote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Establishing Triangles with Different Behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' An interesting test case is the 30-60-90 triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Despite having the spatial equidistribution property from being a rational planar polygon, it is known to be integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' This triangle has pointNineNineTriangle: Percent of dy integrals outside 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' By looking at triangles that are close to the 30-60-90, we can see how sensitive these numerics are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' We ran two runs with a bottom length of .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='575 and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' The bottom length of the 30-60-90 is 1 √ 3 ≈ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' They produced the following results: Triangle ϵ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='01 ϵ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='005 ϵ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='001 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='58 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='6 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='0 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='8 30-60-90 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='7 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='4 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='575 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='8 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='1 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' This complicates our interpretation, as we have a non-ergodic triangle that is displaying behavior that would be expected of an ergodic domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Accuracy and Sanity Checks pointNineNineTriangle: Percent of dy integrals outside 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='001 threshhold 100 90 80 70 09 50 40 30 20 10 0 J 0 200 400 600 800 1000 1200 140020 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' CHRISTIANSON AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' PEZZI (a) 30-60-90 - 450 Nodes - 1000 Eigenvalues (b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='58 triangle - 200 Nodes - 1250 Eigenval- ues 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Mesh Convergence Test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Confidence in our numerics increases if we can show convergence in accuracy metrics as our mesh is refined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' To test this, we chose two metrics: one for the eigenvalue and one for the eigenfunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' The maximum eigenvalue difference is simply the largest difference between eigenvalues computed on the different meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' The L2 running average is the run- ning average of the L2 norm of the difference between the eigenfunctions computed on different meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' To evaluate this difference, the higher accuracy function is interpolated on the coarser mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' This adds another source of inaccuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' We compared the 256 node calculations to the 128, 64, and 32 node calculations for the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='99 triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' The first 1000 eigenvalues and eigenfunctions were computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' The table below shows clear convergence on both metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Comparison Max Eval Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' L2 Running Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 256 and 128 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='87 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='0091 256 and 64 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='23 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='0838 256 and 32 377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='87 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='4762 The 1000th Eigenvalue has a magnitude of around 30,000, so a maximum differ- ence of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='87 corresponds to about a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='03% difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' This shows we are not gaining a substantial amount of accuracy doubling the perimeter node count once we pass 306090Triangle:Percentofdyintegralsoutside0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='01threshhold 1 [ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='1 0 100 200 300 400 500 600 700 800 006 1000pointFiveEightTriangle: Percent of dy integrals outside 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='01 threshhold 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='2 0 200 400 600 800 1000 1200 140021 a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' This gives us confidence that our numerical experiment is well behaved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Reported Errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' FreeFEM itself can also report errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' It does this in 3 types, the relative error, absolute error, the backward error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' All of these errors are generally monotonically increasing, so we will just report the error on the 1250th eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Error Type Value Absolute Error 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='77e-8 Relative Error 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='04e-15 Backwards Error 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='292e-12 Appendices A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Volume and Boundary Data for Near Isosceles Triangles Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='99 Triangle - 1250 Eigenvalues - Y Volume Integral and Bottom Left Neumann plots 99Triangle-YVolumeIntegral-425nodes 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='8 Calcuated Y Volume Integral 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='1 0 0 200 400 600 800 1000 1200 EigenvalueNumber99Triangle-BottomLeftNeumannData-425nodes 0 00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='9 ： 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='8 Data 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='7 O 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='1 0 0 200 400 600 800 1000 1200 EigenvalueNumber22 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='98 Triangle - 1250 Eigenvalues - Y Volume Integral and Bottom Left Neumann plots References [CdV85] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Colin de Verdi`ere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Ergodicit´e et fonctions propres du laplacien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Quantum ergodic restriction theorems, i: interior hypersurfaces in domains with ergodic billiards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Annales Henri Poincar´e, 13:599–670, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' 98Triangle-YVolumeIntegral-200 nodes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content='7 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Quantum ergodic restriction theorems: manifolds with- out boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=', 23(2):715–775, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE1T4oBgHgl3EQf9Qa6/content/2301.03555v1.pdf'} +page_content=' [Zel87] Steven Zelditch.' metadata={'source': 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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: mo_heydari@alumni.iust.ac.ir + + +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. 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Carbotte, "Magneto-optical conductivity in graphene," Journal of Physics: +Condensed Matter, vol. 19, p. 026222, 2006. + + diff --git a/HdAyT4oBgHgl3EQf5frD/content/tmp_files/load_file.txt b/HdAyT4oBgHgl3EQf5frD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a941c793b550126d38d761eac9b81719f4f5df39 --- /dev/null +++ b/HdAyT4oBgHgl3EQf5frD/content/tmp_files/load_file.txt @@ -0,0 +1,1018 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf,len=1017 +page_content='1 New Analytical Approach Based on Transfer Matrix Method (TMM) for Study of Tunable Plasmonic Modes in Graphene-Based Heterostructures Mohammad Bagher Heydari 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Majid Karimipour 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Morteza Mohammadi Shirkolaei 3 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='* School of Electrical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Iran University of Science and Technology (IUST),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Tehran,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Iran 2 Department of Electrical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Arak University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Arak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Iran 3 Department of Electrical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Shahid Sattari Aeronautical University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Tehran,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Iran Corresponding author: mo_heydari@alumni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='iust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='ir 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' An analytical model is presented to derive the reflection characteristics by using the transfer matrix method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The upper Reststrahlen band has been chosen as the studied frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' It is shown that the characteristics of the reflected beam can be effectively controlled by varying the chemical potential of graphene sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The obtained results represent a high value of the reflected group delay (𝜏𝑟 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='3 𝑝𝑠) at the frequency of 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='9 THz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Key-words: Graphene, hBN, plasmon, phonon, analytical 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Introduction Nowadays, graphene has attracted immense interest among scientists in nano-electronics and THz applications [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Graphene-based waveguides have various structures such as planar [16, 24, 33-63], cylindrical [44, 64-69], and elliptical structures [23, 70-72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Combining graphene with other Van der Waals materials can be interesting because hybrid heterostructures have compound properties of both materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The hybridization of graphene with this material can generate and support new types of propagating modes called “coupled phonon-plasmon polaritons” [76-82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' However, in the mid-infrared region, there is limited research related to obtaining large, tunable group delay [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' One of the interesting ways to flexibly obtain and vary the group delay in this band is the usage of hybrid heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Here, we propose a hybrid heterostructure composed of graphene-hBN-graphene layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Two graphene sheets have been utilized in our system to increase the degree of freedom to adjust the reflection characteristics more flexibly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The whole structure is illuminated by a TM-polarized beam with an incident angle of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The analytical expressions 2 are derived for the calculation of the reflection characteristics of the structure by using the transfer matrix method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' A large value of reflected group delay, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 𝜏𝑟 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='3 𝑝𝑠, is achievable at the frequency of 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='9 THz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' A similar study is done in [88], where hyperbolic plasmon- phonon modes are examined by nano-infrared imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' In [92], the reflected group delay is reported as τ=13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='97 ps for the chemical potential of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='25 eV for their structure while our obtained result is about 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='3 ps at the frequency of 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='9 THz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Therefore, our proposed structure is tunable and can change effectively the characteristics of the reflected beam by varying the chemical potential of graphene sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' In section 2, after introducing the proposed structure, mathematical expressions will be presented for the reflected group delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Then, in section 3, the analytical results are reported and investigated more precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' It will be shown that the hybrid structure can flexibly control the reflected beam by changing the chemical potential of graphene sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Finally, section 4 concludes the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The Proposed Heterostructure and its Analytical Model Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The whole structure is illuminated by a TM- polarized beam with an incident angle of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' There is no layer below the Si layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The conductivity of each graphene sheet can be modeled by the following relation [93]: \uf028 \uf029 \uf028 \uf029 ,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 \uf06d \uf06d \uf077 \uf06d \uf073 \uf077 \uf06d \uf070 \uf06d \uf077 \uf070 \uf077 \uf02d \uf0e9 \uf0f9 \uf02d \uf02d \uf047 \uf0e9 \uf0f9 \uf02d \uf02d \uf0ea \uf0fa \uf047 \uf02b \uf02b \uf02b \uf0ea \uf0fa \uf02b \uf02d \uf047 \uf02d \uf047 \uf0ea \uf0fa \uf0eb \uf0fb \uf0eb \uf0fb \uf03d (1) In (1), 𝛤 is the scattering rate, 𝑇 is the temperature, and 𝜇𝑐,1,2 is the chemical potential of each graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The schematic of the studied structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 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]: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 2 2 , , , , 2 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' LO m TO m m m m TO m m j \uf065 \uf065 \uf065 \uf077 \uf077 \uf077 \uf077 \uf077 \uf077 \uf0a5 \uf0a5 \uf03d \uf02b \uf02d \uf02d \uf02d \uf047 (2) In (2), 𝑚 = ‖ 𝑜𝑟 ⊥ is related to the transverse and z-axis, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='9 𝑇𝐻𝑧, 𝜔𝑇𝑂,⊥ = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='4 𝑇𝐻𝑧, 𝜔𝐿𝑂,‖ = 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='3 𝑇𝐻𝑧, 𝜔𝑇𝑂,‖ = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='1 𝑇𝐻𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' In (2), 𝛤𝑚 is a damping factor (𝛤⊥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='15 𝑇𝐻𝑧, 𝛤‖ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='12 𝑇𝐻𝑧) and ɛ𝑚 is related to the high-frequency permittivity (ɛ∞,⊥ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='87, ɛ∞,‖ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='95) [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 2, the dielectric function of hBN permittivity is depicted, which shows the lower and upper Reststrahlen bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The permittivity of hBN versus frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The lower and upper Reststrahlen bands are shown in this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' To calculate the reflection coefficient and the reflected group delay, the transfer matrix method can be utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' For various regions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' TM-polarized waves (p-polarized waves) can be written as follows: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='1 1 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='2 2 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='3 3 3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='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 \uf02d \uf02d \uf02d \uf02d \uf03d \uf02b \uf03c \uf03d \uf02b \uf03c \uf03c \uf03d \uf02b \uf03c \uf03c \uf02b \uf03d \uf02b \uf02b \uf03c (3) Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' the transfer matrix of the whole structure is obtained: 1 4 1 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' a a M b b \uf03d \uf0e9 \uf0f9 \uf0e9 \uf0f9 \uf0ea \uf0fa \uf0ea \uf0fa \uf0eb \uf0fb \uf0eb \uf0fb (4) Where 1 2 2 2 3 3 3 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' M D P D P D \uf0ae \uf0ae \uf0ae \uf03d (5) From Air to hBN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' the transmission matrix is written as follows: 1 2 1 1 1 1 1 2 TM TM TM TM TM TM TM TM D \uf068 \uf078 \uf068 \uf078 \uf068 \uf078 \uf068 \uf078 \uf0ae \uf02b \uf02b \uf02d \uf02d \uf0e6 \uf0f6 \uf03d \uf0e7 \uf0f7 \uf02d \uf02b \uf02b \uf02d \uf0e8 \uf0f8 (6) In (6),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' the following parameters have been defined: 4 0 2 1 z TM z k k \uf065 \uf068 \uf065\uf05e \uf03d (7) 1 2 0 z TM k \uf073 \uf078 \uf065 \uf065 \uf077 \uf05e \uf03d (8) Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 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 \uf071 \uf03d (9) \uf028 \uf029 2 2 2 0 sin z k k \uf065 \uf071 \uf065 \uf065 \uf05e \uf05e \uf03d \uf02d (10) Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 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 \uf02d \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf03d \uf0e7 \uf0f7 \uf0e8 \uf0f8 (11) When the beam inside the hBN medium reaches the surface of the second graphene sheet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 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 \uf068 \uf078 \uf068 \uf078 \uf068 \uf078 \uf068 \uf078 \uf0ae \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf02b \uf02b \uf02d \uf02d \uf0e6 \uf0f6 \uf03d \uf0e7 \uf0f7 \uf0a2 \uf0a2 \uf0a2 \uf0a2 \uf02d \uf02b \uf02b \uf02d \uf0e8 \uf0f8 (12) Where the following parameters have been utilized in (12): 2 3 2 z TM SiO z k k \uf065 \uf068 \uf065 \uf05e \uf0a2 \uf03d (13) 2 2 3 0 z TM SiO k \uf073 \uf078 \uf065 \uf065 \uf077 \uf0a2 \uf03d (14) Similar to the propagation matrix of the beam inside the hBN medium,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 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 \uf02d \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf03d \uf0e7 \uf0f7 \uf0e8 \uf0f8 (15) Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' as the beam reaches the border of SiO2-Si layers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' the transmission matrix can be expressed as: 3 4 1 1 1 1 1 2 TM TM TM TM D \uf068 \uf068 \uf068 \uf068 \uf0ae \uf0a2\uf0a2 \uf0a2\uf0a2 \uf02b \uf02d \uf0e6 \uf0f6 \uf03d \uf0e7 \uf0f7 \uf0a2\uf0a2 \uf0a2\uf0a2 \uf02d \uf02b \uf0e8 \uf0f8 (16) In (16),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' the following parameters have been defined: 2 4 3 SiO z TM Si z k k \uf065 \uf068 \uf065 \uf0a2\uf0a2 \uf03d (17) By calculating the elements of the transfer matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' the reflection coefficient and the reflectance are derived by: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 21 11 exp r M r r j M \uf077 \uf06a \uf077 \uf03d \uf03d (18) \uf028 \uf029 2 R r \uf077 \uf03d (19) For the proposed structure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 𝑀21 and 𝑀11 are calculated: 5 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 11 3 2 3 2 3 2 3 2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 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 \uf068 \uf078 \uf068 \uf078 \uf068 \uf068 \uf078 \uf068 \uf078 \uf068 \uf068 \uf078 \uf068 \uf078 \uf068 \uf068 \uf078 \uf068 \uf078 \uf068 \uf02d \uf02d \uf02d \uf02b \uf02b \uf02d \uf02b \uf02b \uf0a2 \uf0a2 \uf0a2\uf0a2 \uf03d \uf02b \uf02b \uf02b \uf02b \uf02b \uf02b \uf0a2 \uf0a2 \uf0a2\uf0a2 \uf02d \uf02d \uf02d \uf02b \uf02b \uf02b \uf0a2 \uf0a2 \uf0a2\uf0a2 \uf02b \uf02b \uf02d \uf02d \uf02d \uf02b \uf0a2 \uf0a2 \uf0a2\uf0a2 \uf02d \uf02d \uf02b \uf02d \uf02d (20) \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 21 3 2 3 2 3 2 3 2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 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 \uf068 \uf078 \uf068 \uf078 \uf068 \uf068 \uf078 \uf068 \uf078 \uf068 \uf068 \uf078 \uf068 \uf078 \uf068 \uf068 \uf078 \uf068 \uf078 \uf068 \uf02d \uf02d \uf02d \uf02b \uf02b \uf02d \uf02b \uf02b \uf0a2 \uf0a2 \uf0a2\uf0a2 \uf03d \uf02d \uf02b \uf02b \uf02b \uf02b \uf02b \uf0a2 \uf0a2 \uf0a2\uf0a2 \uf02b \uf02d \uf02d \uf02b \uf02b \uf02b \uf0a2 \uf0a2 \uf0a2\uf0a2 \uf02d \uf02b \uf02d \uf02d \uf02d \uf02b \uf0a2 \uf0a2 \uf0a2\uf0a2 \uf02b \uf02d \uf02b \uf02d \uf02d (21) Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' if we suppose that the incident pulse is a Gaussian beam with the central and half-width of 𝜔0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 𝜏0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' respectively: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 2 2 0 0 0 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' exp 2 exp iE t A t i t \uf074 \uf077 \uf03d \uf02d \uf02d (22) It should be noted that the corresponding Fourier spectrum of (22) is: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 2 2 2 0 0 0 0 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' exp 2 2 i A E \uf074 \uf077 \uf077 \uf077 \uf074 \uf070 \uf03d \uf02d \uf02d (23) and thus the group delay is not a function of time (t) because all relations are written in the Fourier space (ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Then, the reflected group delay is obtained: \uf028 \uf029 r r c \uf077 \uf077 \uf06a \uf077 \uf074 \uf077 \uf03d \uf0b6 \uf0e9 \uf0f9 \uf03d \uf0ea \uf0fa \uf0b6 \uf0eb \uf0fb (24) In (24), 𝜔𝑐 is the carrier frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Now, our model is completed for the proposed heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' In what follows, we will investigate the analytical results of the above mathematical relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Results and Discussions This section reports the analytical results of the proposed structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' In these results, the chemical potential of graphene sheets is supposed to be 𝜇𝑐,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='2 𝑒𝑣, 𝜇𝑐,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='3 𝑒𝑣, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The temperature is 𝑇 = 300 𝐾 and the relaxation time is assumed to be 𝜏 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='45 𝑝𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Both graphene layers have the similar thickness 𝛥1 = 𝛥2 = 𝛥 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='33 𝑛𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The parameters of the hBN medium have been given in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Moreover, the geometrical parameters are 𝑡 = 100𝑛𝑚, 𝑑 = 150𝑛𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The permittivity constant of SiO2 and Si layers are assumed to be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='09 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='9, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The incident angle is 𝜃 = 450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' As explained before, there are two phonon modes in the hBN medium (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 2) that are related to hyperbolicity: out-of-plane and in-plane phonon modes, which lead to two various Reststrahlen bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' First, let us consider the reflected group delays in three different frequency ranges: 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='75-25 THz (in the lower Reststrahlen band), 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='15-37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='5 THz (in the middle band, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' between the lower and upper Reststrahlen bands) and 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='1-41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='6 THz (in the upper Reststrahlen band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' It should be emphasized that the resonance frequency at 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='855 THz in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 3 (a) is not related to the lower TO phonon frequency (𝜔𝑇𝑂,⊥ = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='4 𝑇𝐻𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' As observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 3 (a), the group delay shows the metal-like behavior in the lower Reststrahlen band which can be enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' While the reflected group delay in other frequency ranges (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Therefore, we only focus on the first frequency window, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='15-25 THz (the lower Reststrahlen band), in the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' It should be noted that the reflected group delay in our structure originates from the Lorentz resonance mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 6 Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Reflected group delay versus frequency at three frequency windows: (a) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='75 THz-25 THz (in the lower Reststrahlen band), (b) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='15 THz-37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='5 THz (in the middle band, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' between the lower and upper Reststrahlen bands), (c) 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='1 THz-41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='6 THz (in the upper Reststrahlen band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The chemical potential of graphene layers is supposed to be 𝜇𝑐,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='2 𝑒𝑣, 𝜇𝑐,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='3 𝑒𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The thickness of the hBN and SiO2 layers are 100 nm and 150 nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The incident angle is 𝜃 = 450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' In the previous section, we analytically obtained the elements of the transfer matrix for the proposed heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The studied structure is a tunable device in which its reflection characteristics can be varied by changing the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 4 represents the variations of reflectance, the group delay, and the reflected phase as a function of frequency for various values of chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' As noted before, the frequency range is 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='75-25 THz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The incident angle is chosen 𝜃 = 450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' It can be found from fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 4 (a) that the reflectance has a dip of around 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='88 THz and its place can be varied as the values of chemical potential change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Around 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='88 THz, the sign of reflected phase changes, as seen in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 4 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Meanwhile, the peak of the reflected group delay varies for various values of chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' A large value of reflected group delay, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 𝜏𝑟 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='2 𝑝𝑠, is reported for the chemical potentials of 𝜇𝑐,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='2 𝑒𝑣, 𝜇𝑐,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='8 𝑒𝑣 at the frequency of 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='85 THz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 5 shows the reflected group delay of the optical beam as a function of frequency for various values of graphene thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' In this diagram, it is supposed that both graphene sheets have similar thicknesses (𝛥1 = 𝛥2 = 𝛥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The chemical potential of graphene layers are remained fixed: 𝜇𝑐,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='2 𝑒𝑣, 𝜇𝑐,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='3 𝑒𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' As the number of graphene layers increases (the thickness increases), the peak value of the group delay increases, as observed in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Furthermore, one can see that the maximum peak shifts to the higher frequencies as the thickness increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' For thicker graphene sheets, a high value of group delay is achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' For instance, the reflected group delay of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='3ps is obtained for the thickness of 𝛥1 = 𝛥2 = 𝛥 = 1 𝑛𝑚 at the frequency of 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='9 THz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' One of these parameters is the thickness of the hBN layer, where its variations have been depicted in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The chemical potential of graphene layers are 𝜇𝑐,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='2 𝑒𝑣, 𝜇𝑐,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='3 𝑒𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Both graphene layers have the similar thickness 𝛥1 = 𝛥2 = 𝛥 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='33 𝑛𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Other parameters have remained fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' It 7 can be seen from fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 5 that the maximum point of the reflected group delay shifts to higher frequencies as the thickness of the hBN medium increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' However, the changes are slight because the thickness of the hBN layer varies from 100nm to 120 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The thickness of the hBN and SiO2 layers are 100 nm and 150 nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The incident angle is 𝜃 = 450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The reflected group delay versus frequency for different values of graphene thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' It is supposed that both graphene layers have similar thicknesses (𝛥1 = 𝛥2 = 𝛥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The chemical potential of graphene layers is supposed to be 𝜇𝑐,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='2 𝑒𝑣, 𝜇𝑐,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='3 𝑒𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The thickness of the hBN and SiO2 layers are 100 nm and 150 nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The incident angle is 𝜃 = 450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' As a final point, we investigate the influence of SiO2 thickness on the reflected group delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' As derived in relations (16)-(21), the thickness of the SiO2 layer can change the characteristics of the reflected beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' One can observe from fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 7 that as the SiO2 thickness varies from 150nm to 200 nm, the maximum peak changes, and its frequency shifts to higher frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 8 Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The reflected group delay versus frequency for different values of hBN thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' It is supposed that both graphene layers have similar thicknesses (𝛥1 = 𝛥2 = 𝛥 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='33 𝑛𝑚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The chemical potential of graphene layers is supposed to be 𝜇𝑐,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='2 𝑒𝑣, 𝜇𝑐,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='3 𝑒𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The thickness of the SiO2 layer is 150 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The incident angle is 𝜃 = 450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The reflected group delay versus frequency for different values of SiO2 thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' It is supposed that both graphene layers have similar thicknesses (𝛥1 = 𝛥2 = 𝛥 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='33 𝑛𝑚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The chemical potential of graphene layers is supposed to be 𝜇𝑐,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='2 𝑒𝑣, 𝜇𝑐,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='3 𝑒𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The thickness of the hBN layer is 100 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The incident angle is 𝜃 = 450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Conclusion In this article, we studied the characteristics of the reflected beam from graphene-based hBN heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Analytical expressions were obtained for calculating the reflection characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' A large value of the reflected group delay was seen in the lower Reststrahlen band;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' therefore, this frequency range was chosen to be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Our results reported a large value of the reflected group delay, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 𝜏𝑟 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='3 𝑝𝑠, at the frequency of 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content='9 THz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' The authors believe that the presented study can be utilized for the design of optical delay structures in the mid-infrared region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Declarations Ethics Approval: Not Applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Consent to Participate: Not Applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Consent for Publication: Not Applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Funding: The authors received no specific funding for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Conflicts of Interest/ Competing Interests: The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' 9 Availability of Data and Materials: Not Applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Code availability: Not Applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=" Authors' Contributions: M." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Heydari proposed the main idea of this work and performed the analytical modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' Karimipour conducted the numerical simulations and wrote the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQf5frD/content/2301.00807v1.pdf'} +page_content=' M.' 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0000000000000000000000000000000000000000..3502bfbc58801cdf36f56614e036b109aef6c7a3 --- /dev/null +++ b/HtE2T4oBgHgl3EQfTwfs/content/tmp_files/2301.03807v1.pdf.txt @@ -0,0 +1,2092 @@ +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 +[1] Agore, A.L. - Functors between representation categories. Universal modules, arXiv:2301.03051. 10 +[2] Agore, A.L., Gordienko, A.S., Vercruysse, J. - V -universal Hopf algebras (co)acting on Ω-algebras, +Commun. Contemp. Math. 25 (2023), 2150095. 2, 19 +[3] Agore, A.L. - Universal coacting Poisson Hopf algebras, Manuscripta Math. 165 (2021), 255–268. 2 +[4] Agore, A.L., Gordienko, A.S., Vercruysse, J. - Equivalences of (co)module algebra structures over +Hopf algebras, J. Noncommut. Geom. 15 (2021), 951–993. 3 +[5] Agore, A.L., Militaru, G. - A new invariant for finite dimensional Leibniz/Lie algebras, J. Algebra +562 (2020), 390–409. 3, 15, 17, 18, 19, 20 +[6] Agore, A.L., Militaru, G. - Jacobi and Poisson algebras, J. Noncommut. Geom., 9 (2015), 1295–1342. +4, 9 +[7] Ardizzoni, A., El Kaoutit, L., Menini, C. - Coendomorphism left bialgebroids, J. Algebra and Its +App., 12 (2013), No. 03, 1250181. 2, 3 +[8] Ardizzoni, A., El Kaoutit, L., Menini, C. - Categories of comodules and chain complexes of modules, +Intern. J. Math., 23 (2012), No. 10, 1250109. 2, 3 +[9] Ballesteros, A., Herranz, F.J., Musso, F., Ragnisco, O. - Superintegrable deformations of the +Smorodinsky-Winternitz Hamiltonian, Superintegrability in classical and quantum systems, CRM +Proc. Lecture Notes 37, Amer. Math. Soc., Providence, RI, 2004. 5 +[10] Bhattacharjee, S., Chirvˇasitu, A., Goswami, D. - Quantum Galois groups of subfactors, Internat. J. +Math. 33 (2022), 2 (2022) 2250013. 3 + +22 +A. L. AGORE AND G. MILITARU +[11] Chirvˇasitu, A., Walton, C., Wang, X. - On quantum groups associated to a pair of preregular forms, +J. Noncommut. Geom. bf 13 (2019), 115—159. 3 +[12] Crainic, M., Fernandes, R. L., Mˇarcut¸, I. – Lectures on Poisson geometry, Graduate Studies in +Mathematics, 217 (2021), Amer. Math. Soc., Providence. 1, 5 +[13] Drinfeld, V. G. – Quantum groups, Proceedings of the 1986 International Congress of Mathematics, +1 (1987) 798–820. 1 +[14] Gu´ed´enon, T. - Fundamental Theorem of Poisson (A, H)-Hopf modules, J. Algebra, 595, (2022), +216–243. 3, 5 +[15] Grabowski, J. - Brackets, Int. J. Geom. Methods Mod. Phys., 10(8):1360001, 45, 2013. 1 +[16] Huang, H., Walton, C., Wicks, E., Won, R. - Universal quantum semigroupoids, J. Pure Appl. +Algebra 227 (2023), 107193. 3 +[17] Huang, H., Van Nguyen, C., Ure, C., Vashaw, K.B., Veerapen, P., Wang, X. - Twisting Manin’s +universal quantum groups and comodule algebras, arXiv:2209.11621 3 +[18] Huebschmann, J. - Poisson cohomology and quantization, J. Reine Angew. Math., 408 (1990), +57–113. 1 +[19] Kontsevich, M. - Deformation Quantization of Poisson Manifolds, Letters of Mathematical Physics, +66 (2003), 157–216. 1 +[20] Laurent-Gengoux, C., Pichereau, A., Vanhaecke, P. – Poisson Structures, Vol. 347, 2013, Springer. +1, 5 +[21] Mac Lane, S. - Categories for the Working Mathematician, Graduate Texts in Mathematics, 5 +(Second ed.), Springer, 1998, ISBN 0-387-98403-8. 5 +[22] Manin, Yu. I. - Quantum groups and noncommutative geometry, Universite de Montreal, Centre de +Recherches Mathematiques, Montreal, QC, 1988. 1, 2 +[23] Militaru, G. - The automorphisms group and the classification of gradings of finite dimensional +associative algebras, Results Math. 77 (2022). 2, 3, 19, 20 +[24] Takeuchi, M. - Free Hopf algebras generated by coalgebras, J. Math. Soc. Japan 23 (1971), 561–582. +18 +[25] Tambara, D. - The coendomorphism bialgebra of an algebra. J. Fac. Sci. Univ. Tokyo Math. 37 +(1990), 425–456. 1, 2, 6 +[26] Radford, D.E. - Hopf algebras, World Scientific, 2012. 5, 19 +[27] Raedschelders, T., Van den Bergh. M. - The Manin Hopf algebra of a Koszul Artin–Schelter regular +algebra is quasi-hereditary, Adv. Math., 305 (2017), 601–660. 2 +[28] Sweedler, M.E. - Hopf Algebras, Benjamin New York, 1969. 1, 2, 4, 5 +[29] Umirbaev, U. - Universal enveloping algebras and universal derivations of Poisson algebras, J. Al- +gebra, 354 (2012), 77–94. 4 +[30] Van den Bergh, M. - Double Poisson algebras, Trans. Amer. Math. Soc., 360 (2008), 5711–5769. 1 +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: ana.agore@gmail.com +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: gigel.militaru@fmi.unibuc.ro and gigel.militaru@gmail.com + diff --git a/HtE2T4oBgHgl3EQfTwfs/content/tmp_files/load_file.txt b/HtE2T4oBgHgl3EQfTwfs/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..17510b01d820adc4ef8a7c05830d3a5951e258b2 --- /dev/null +++ b/HtE2T4oBgHgl3EQfTwfs/content/tmp_files/load_file.txt @@ -0,0 +1,779 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf,len=778 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='03807v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='RA] 10 Jan 2023 UNIVERSAL CONSTRUCTIONS FOR POISSON ALGEBRAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' APPLICATIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' AGORE AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' MILITARU Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' The universal algebra is shown to exist for any finite dimensional Poisson algebra P and several of its applications are highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Two applications are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' C∞(M, R) is a Poisson algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' In particular, the existence and description of universal objects (groups 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 17B63, 16T05, 16T10, 16W20, 16W50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Poisson algebras, universal constructions, automorphisms group, gradings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 1 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' AGORE AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' MILITARU or ”group like objects” such as Lie groups, algebraic groups, Hopf algebras, groupoids or quantum groupoids, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=') which act or coact on a fixed object O in a certain category C has often various applications in many areas of mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Then the category ActGrO has a final object, namely � AutC(O), Id � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=') 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Indeed, the first obstacle we encounter is the fact that AutC(O) might not be an object inside the category D anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' To the best of our knowledge, the first result in this direction was proved by Sweelder [28, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' A is an H-module algebra) has a final object M(A), called by Sweedler the universal measuring bialgebra of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' If A is an associative algebra, necessarily finite dimensional this time around, then the category of all bialgebras that coact on A (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' A is an H-comodule algebra) has an intial object a(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' The results have been extended in recent years to the setting of bialgebroids coactions in [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='1] and a(A)o is just Sweedler’s final object in the category of all bialgebras that act on A [25, Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' The existence and description of these universal (co)acting bialgebras/Hopf algebras has been considered recently in [2] in the context of Ω-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Furthermore, universal coacting objects for Poisson algebras have also been considered in [3] but from a different perspective, leading to entirely different constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' We only point out that in [3], the universal coacting object considered is actually a Poisson Hopf algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' This result has two important consequences: as proved in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='5, for a fixed Poisson P-module U there exists a canonical functor U ⊗ −: AM → QPM from the category of usual A-modules (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' representations of the associative algebra A) to the category of Poisson Q-modules (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Poisson repre- sentations of Q) and moreover, if U is finite dimensional this functor has a left adjoint (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' In Section 3 we consider three more applications of our constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' P(P) has a canonical bial- gebra structure and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' (2) Describe and classify all G-gradings on P for a given abelian group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' More precisely, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' The second application is given in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' It is reasonable to hope that H(P) will play the role of a non-commutative symmetry group of the Poisson algebra P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' AGORE AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' MILITARU 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Throughout, δs,1 will stand for Kronecker’s symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' (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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' if P1 and P2 are unital Poisson algebras then a Poisson homomorphism will be assumed to preserve units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' We denote by AutPoiss(P) the automorphisms group of a Poisson algebra P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Let P be a Poisson algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Remarks 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' A representation of a Poisson algebra P on a vector space V [6, Remarks 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' The concepts of a Poisson P-module structure on V and a representation of P on V are obviously equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' We shall denote by Poissk, Poiss1 k and ComAlgk the categories of Poisson, unital Poisson and respectively unital commutative associative algebras over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Furthermore, the cat- egory of commutative bialgebras (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Hopf algebras) is denoted by ComBiAlgk (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' ComHopfk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' For a coalgebra C we denote by G(C) the set of group like elements of C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' G(C) := {x ∈ C | ∆(x) = x ⊗ x and ε(x) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' If B is a bialgebra, then G(B) is a monoid with respect to the multiplication on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Throughout, for a bialgebra B, we denote by Bo its finite dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' For a fixed Poisson algebra P we denote by CoactBialgP (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' CoactHopfP ) the category of all commutative bialgebras (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Hopf algebras) coacting on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' That is, the objects are all pairs (H, ρP ) consisting of a commutative bialgebra (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Examples 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' The first basic example of a Poisson H-comodule algebra is the one induced by G-graded Poisson algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Let k[G] be the group algebra of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' By extending a well known result in Hopf algebra theory ([26, Excercise 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' The bijection is given such that xσ ∈ Pσ if and only if ρ(xσ) = xσ ⊗ σ, for all σ ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' The second example of a Poisson comodule algebra comes from algebraic geometry [14, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='20]: if V is an affine Poisson variety (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' For further details concerning the study of Poisson algebras see [12, 20] and the references therein and for undefined concepts on category theory (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Hopf algebras) we refer the reader to [21] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' [26, 28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' (4) 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' AGORE AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' (1) holds for P ⊗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Furthermore, if f : A → B is an algebra map then IdP ⊗ f : P ⊗ A → P ⊗ B is a morphism of Poisson algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' With this remark in hand we can now introduce the following concept: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Let P and Q be two Poisson algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' g = � IdP ⊗ θ � η (5) If Q = P then P(P) := P(P, P) will be called the universal coacting bialgebra on P 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' The universal algebra of two Poisson algebras P and Q, if exists, it is unique up to an isomorphism of algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' We formulate this result in terms of adjoint functors, as the Poisson algebra version of [25, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Let P be a finite dimensional Poisson algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Then the functor P ⊗ − : ComAlgk → Poissk has a left adjoint P(P, −) : Poissk → ComAlgk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Furthermore, if Q is an arbitrary Poisson algebra, then P(P, Q) is the universal algebra of P and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Let n ∈ N∗ be a positive integer and {e1, · · · , en} a basis of the Poisson algebra P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' (6) We will construct explicitly a left adjoint P(P, −) : Poissk → ComAlgk for the tensor product functor P ⊗ − : ComAlgk → Poissk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' To this end, let Q be a Poisson algebra 1The terminology is explained by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' UNIVERSAL CONSTRUCTIONS FOR POISSON ALGEBRAS 7 and consider {fi | i ∈ I} to be its basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Next, we consider the following linear map: ηQ : Q → P ⊗ P(P, Q), ηQ(fi) := n � s=1 es ⊗ xsi, for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' (12) We will see that ηQ is in fact a Poisson algebra map;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' AGORE AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' We have to prove that there exists a unique algebra homomorphism θ: P(P, Q) → A such that g = � IdP ⊗ θ � ηQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Let {dsi | s = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' as g: Q → P ⊗ A is a Poisson algebra map,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' we can easily conclude that the following compatibilities hold for all a = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' n and i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' j ∈ I: � u∈Ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='j αu i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='j dau = n � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='t=1 τ a s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='t dsidtj (15) � u∈Vi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='j βu i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='j dau = n � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='t=1 µa s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='t dsidtj (16) The universal property of the polynomial algebra yields a unique algebra homomorphism v: k[Xsi |s = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' i ∈ I] → A such that v(Xsi) = dsi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' for all s = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' n and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Furthermore, we have J ⊆ Ker(v), where J is the ideal generated by all polynomials listed in (8) and (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' for all i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' j ∈ I and a = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' n we have: v � Γ(P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Q) (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='j) � = v � � u∈Ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='j αu i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='j Xau − n � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='t=1 τ a s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='t XsiXtj � = � u∈Ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='j αu i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='j dau − n � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='t=1 τ a s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='t dsidtj (15) = 0 v � Ω(P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Q) (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='j) � = v � � u∈Vi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='j βu i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='j Xau − n � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='t=1 µa s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='t XsiXtj � = � u∈Vi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='j βu i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='j dau − n � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='t=1 µa s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='t dsidtj (16) = 0 Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' there exists a unique algebra homomorphism θ: P(P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Q) → A such that θ(xsi) = dsi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' for all s = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' n and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' We are left to show that g = � IdP ⊗ θ � ηQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' We are left to show that θ is the unique morphism with this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Indeed, consider ˜θ: P(P, Q) → A to be another algebra homomorphism such that � IdP ⊗ ˜θ � ηQ(fi) = g(fi), for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' As the set {xsi |s = 1, · · · , n, i ∈ I } generates the algebra P(P, Q) we can conclude that ˜θ = θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' To summarize, we proved that the map γQ, A given by (13) is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Indeed, let u: Q1 → Q2 be a Poisson algebra homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Moreover, it can now be easily checked that P(P, −) is indeed a functor and that γQ, A is natural in both variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Therefore, the functor P(P, −) is the left adjoint of the functor P ⊗ −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Finally, the bijectivity of the map (13) shows that the pair � P(P, Q), ηQ � is indeed the universal algebra of P and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='2 remains valid if we replace Poissk by the category Poiss1 k of unital Poisson algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' These new relations are necessary and sufficient for the map ηQ: Q → P ⊗ P1(P, Q) defined in (12) to be unital, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' ηQ(1Q) = 1P ⊗1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' The rest of the proof goes exactly as for Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Furthermore, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='2 can be generalized to the category of Jacobi algebras by repeating verbatim the above proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' The universal algebra P(P, Q) of two Poisson algebras P and Q as constructed in Theo- rem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Indeed, by considering A := k, the bijection described in (13) comes down to the following: 10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' AGORE AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' MILITARU Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Let P and Q be two Poisson algebras such that P is finite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Poisson modules) of the two Poisson algebras P and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' similarly, any fixed (V, ·) ∈ AM gives rise to a functor − ⊗ V : PPM → QPM connecting the categories of Poisson modules over P and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' We start by showing that (U ⊗ V, ⊲) is a left Q-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' To thie end,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' we have: (fifj) ⊲ (l ⊗ t) (7) = � u∈Ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='j αu i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='jfu ⊲ (l ⊗ t) (20) = � u∈Ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='r=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='n (αu i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='jer ◮ l) ⊗ (xru · t) = n � r=1 (er ◮ l) ⊗ � � u∈Ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='j αu i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='jxru � t (10) = n � r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='p=1 τ r s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='p (er ◮ l) ⊗ (xsixpj) · t = n � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='p=1 � n � r=1 τ r s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='p er � ◮ l ⊗ (xsixpj) · t (6) = n � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='p=1 (esep) ◮ l ⊗ (xsixpj) · t = n � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' ⇀) being a left Lie Q-module can be proved exactly as in (the proof of) [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' The proof will be finished once we prove that compatibilities UNIVERSAL CONSTRUCTIONS FOR POISSON ALGEBRAS 11 (2) and (3) hold for (U ⊗ V, ⊲, ⇀).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' as compatibilities (2) and (3) hold for (U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' ◮,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' ↷) and A is a commutative algebra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' for all i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' j ∈ I and l ∈ U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' t ∈ V ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' we have: (fifj) ⇀ (l ⊗ t) (7) = � u∈Ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='j αu i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='jfu ⇀ (l ⊗ t) (21) = � u∈Ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='r=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='n (αu i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='jer ↷ l) ⊗ (xru · t) = n � r=1 (er ↷ l) ⊗ � � u∈Ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='j αu i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='jxru � t (10) = n � r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='p=1 τ r s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='p (er ↷ l) ⊗ (xsixpj) · t = n � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='p=1 � n � r=1 τ r s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='p er � ↷ l ⊗ (xsixpj) · t (6) = n � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='p=1 (esep) ↷ l ⊗ (xsixpj) · t (2) = n � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='p=1 � es ◮ (ep ↷ l) + ep ◮ (es ↷ l) � ⊗(xsixpj) · t = n � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='p=1 es ◮ (ep ↷ l) ⊗ xsi · (xpj · t) + n � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' fj] ⊲ (l ⊗ t) (7) = � v∈Vi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='j βu i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='j fu ⊲ (l ⊗ t) (20) = � u∈Vi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='r=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='n βu i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='j(er ◮ l) ⊗ (xru · t) = n � r=1 (er ◮ l) ⊗ � � u∈Vi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='j βu i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='jxru � t (11) = n � r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='p=1 µr s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='p (er ◮ l) ⊗ (xsixpj) · t = n � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='p=1 � n � r=1 µr s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='p er � ◮ l ⊗ (xsixpj) · t (6) = n � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='p=1 [es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' ep] ◮ l ⊗ (xsixpj) · t (3) = n � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='p=1 � es ↷ (ep ◮ l) − ep ↷ (es ◮ l) � ⊗ (xsixpj) · t = n � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='p=1 � es ↷ (ep ◮ l) � ⊗ xsi · (xpj · t) − n � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' □ 12 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' AGORE AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' MILITARU Furhermore, if (U, ◮, ↷) ∈ PPM is finite dimensional then the first functor constructed in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='5 admits a left adjoint: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Then the functor U ⊗ −: AM → QPM has a left adjoint U(U, −): QPM → AM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' The left adjoint U(U, −): QPM → AM of the tensor product functor U ⊗ − will be constructed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' First, consider (V, ⊢, ↬) ∈ QPM and {vr | r ∈ J} its k-basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Consider now the following linear map: ρV : V → U ⊗ U(U, V ), ρV (vr) := m � s=1 us ⊗ ysr, for all r ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' (28) UNIVERSAL CONSTRUCTIONS FOR POISSON ALGEBRAS 13 Note that ρV is a Poisson Q-module map;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' for all j ∈ I and i ∈ J we have: ρV (fj ⊢ vi) (23) = ρV � � p∈Wj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='i σp ji vp � = � p∈Wj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='i m � s=1 σp ji us ⊗ ysp = m � s=1 � us ⊗ � p∈Wj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='i σp ji ysp � (26) = m � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='t=1 n � r=1 γs r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='t us ⊗ xrj ⋄ yti = m � t=1 n � r=1 � m � s=1 γs r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='i ηp ji vp � = � p∈Tj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='i m � s=1 ηp ji us ⊗ ysp = m � s=1 � us ⊗ � p∈Tj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='i ηp ji ysp � (27) = m � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='t=1 n � r=1 ωs r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='t us ⊗ xrj ⋄ yti = m � t=1 n � r=1 � m � s=1 ωs r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' To this end, let g: V → U ⊗ X be a Poisson Q-module map;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' we need to find a unique A-module map θ: U(U, V ) → X such that g = � IdU ⊗θ � ρV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Moreover, Ker(θ) contains the A-submodule of U(U, V ) generated 14 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' AGORE AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' MILITARU by the elements listed in (24) and (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Indeed, let h: V1 → V2 be a Poisson Q-modules map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Moreover, it can now be easily checked that U(U, −) is indeed a functor and that ΓV, X is natural in both variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Therefore, U(U, −) is the left adjoint of the functor U ⊗−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' □ Keeping the notations and the assumptions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='5 we can prove the following: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Then the functor − ⊗ V : PPM → QPM has a left adjoint V(V, −): QPM → P PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Since the proof goes in the same manner as the one of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='6, we only indicate its main steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Using Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' The proof is now finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' □ 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Consequently, relations (11) are automatically fulfilled for i = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Examples 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' xy := 0, for any x, y ∈ P (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Thus P and Q are just Lie algebras viewed as Poisson algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Then, P(P, Q) is exactly the universal algebra A(P, Q) of the two Lie algebras as constructed in [5, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Let P := k be the 1-dimensional Poisson algebra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' the constant structures are τ 1 1,1 = 1 and µ1 1,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 16 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' AGORE AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' The conclusion now follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' The universal coacting bialgebra on a finite dimensional Poisson algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Applications Let P be a finite dimensional Poisson algebra having {e1, · · · , en} as a k-basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' The description of the commutative algebra P(P) := P(P, P) given by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' By considering Q := P in the bijection described in (13) we obtain: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Let P be a finite dimensional Poisson algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' UNIVERSAL CONSTRUCTIONS FOR POISSON ALGEBRAS 17 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Let P be a Poisson algebra of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Consider the Poisson algebra homomorphism (ηP ⊗IdP(P )) ◦ ηP : P → P ⊗P(P)⊗ P(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='1 yields a unique algebra homomorphism ∆ : P(P) → P(P) ⊗ P(P) such that the following holds: (IdP ⊗ ∆) ◦ ηP = (ηP ⊗ IdP(P )) ◦ ηP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Note that ∆ is obviously coassociative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' In a similar fashion, applying once again Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Applying (39) for each ei, i = 1, · · · , n, we obtain ε(xij) = δi,j, for all i, j = 1, · · · , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' It can be easily checked that ε is a counit for ∆, and therefore P(P) is a bialgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Furthermore, (38) and (39) imply that the canonical map ηP : P → P ⊗ P(P) defines a right P(P)-comodule structure on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' □ The key property of P(P) is the following Poisson algebra version of [5, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='11]: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Let P be a finite dimensional Poisson algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' AGORE AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' MILITARU To start with, using Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='1, we obtain a unique algebra homomorphism θ: P(P) → B such that diagram (40) commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' The proof will be finished once we show that θ is a coalgebra homomorphism as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' This follows by using again Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' The proof will be finished once we show that (θ ⊗ θ) ◦ ∆: P(P) → B ⊗ B fulfills the same compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Similarly, one can show that εB ◦ θ = ε and the proof is now finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' □ By considering Takeuchi’s commutative Hopf envelope [24] of the bialgebra P(P) we obtain, using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='3, the following: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Let P be a finite dimensional Poisson algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' The dual versions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='3 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='4 regarding the actions of commutative bialgebras (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Hopf algebras) on a Poisson algebra also hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' For a Poisson algebra P, we can define the category ActBialgP (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' ActHopfP ) of all commutative bialgebras (respectively Hopf algebras) which act on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' More precisely, the objects of ActBialgP (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' ActHopfP) are pairs (B, µP) consisting of a commutative bialgebra (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' µP is a (right) B-module structure on P as well as a Poisson algebra map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Using the same arguments as in [2, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='14], we can prove that ActBialgP (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' ActHopfP ) has a final object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Next we will present two important applications of the bialgebra P(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' These are the Poisson algebra version of similar results obtained for Lie/associative algebras in [5, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 62])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Using Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' (θ1 ⋆ θ2)(xsj) = n � t=1 θ1(xst)θ2(xtj) (43) for all θ1, θ2 ∈ G � P(P)o� and j, s = 1, · · · , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' □ Next, for a given abelian group G, we describe all G-gradings on a Poisson algebra P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Let G be an abelian group and P a finite dimensional Poisson algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' The proof is now finished since we have shown in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='2 that the latter set is in bijective correspondence with the set of all G-gradings on the Poisson algebra P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' □ 20 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' AGORE AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' MILITARU Our next aim is to classify all G-gradings on a Poisson algebra P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' To this end, we introduce the following: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Let G be an abelian group and P a finite dimensional Poisson algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' The next theorem classifies all G-gradings on a Poisson algebra P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Since the associative and Lie/Leibniz algebra counterparts of this result have been proved in detail in [23, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='4] and [5, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='5], respectively, we will be brief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' First, note that by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Indeed, recall from Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' □ 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' The conclusion now follows by denoting � X = x11, �Y = x21, �Z = x31 and �T = x33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' References [1] Agore, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' - Functors between representation categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content=' Universal modules, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='03051.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='unibuc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='ro and gigel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='militaru@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE2T4oBgHgl3EQfTwfs/content/2301.03807v1.pdf'} diff --git a/I9E4T4oBgHgl3EQfhQ2p/content/tmp_files/2301.05124v1.pdf.txt b/I9E4T4oBgHgl3EQfhQ2p/content/tmp_files/2301.05124v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..dfce73b5e9c90b1cf79b4047924ffa6698d44a60 --- /dev/null +++ b/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, stefanie.schneider@itg.uni-muenchen.de; Ricarda Vollmer, ricarda.vollmer@campus.lmu.de, 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. +REFERENCES +[1] Karl von Amira. 1905. Die Handgebärden in den Bilderhandschriften des Sachsenspiegels. Abhandlungen der Bayerischen Akademie +der Wissenschaften. 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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 + diff --git a/I9E4T4oBgHgl3EQfhQ2p/content/tmp_files/load_file.txt b/I9E4T4oBgHgl3EQfhQ2p/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f67f9077b55239990293ad6592805d387b94f2c4 --- /dev/null +++ b/I9E4T4oBgHgl3EQfhQ2p/content/tmp_files/load_file.txt @@ -0,0 +1,1834 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf,len=1833 +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' among these are mainly joints such as elbows and knees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Each image annotation, in addition to mandatory fields, provides metadata from the art-historical online encyclopedia WikiArt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' CCS Concepts: • Information systems → Recommender systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Image search;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' • Computing methodologies → Object detection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Interest point and salient region detections;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' • Applied computing → Fine arts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' This particularly applies to the drawing apprenticeship [61], whose best-known example is Leonardo da Vinci’s Vitruvian Man.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=', by the Physiognomist Johann Caspar Lavater [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=', 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Authors’ address: Stefanie Schneider, stefanie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='schneider@itg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='uni-muenchen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='de;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Ricarda Vollmer, ricarda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='vollmer@campus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='lmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='de, Ludwig Maximilian University of Munich, Geschwister-Scholl-Platz 1, Munich, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='05124v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='CV] 12 Jan 2023 2 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Schneider and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Vollmer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' We differentiate between two annotation modes: bounding box and keypoint annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' wrists or knees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=', Convolutional Neural Networks (CNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Existing data sets fall broadly into two categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Or they are freely accessible to the public but enclose human figures only by rectangular bounding boxes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' their pose is then broadly categorized without specifically delineating keypoints [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Our contributions are three-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 1 illustrates both annotation modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Pose may thus emerge as wholly elemental to the formulaic recapitulation of significant topoi and motifs through computational assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' (iii) As a by-product of PoPArt’s domain-specific curation, the sole detection of figures in art-historical collections is decisively improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' It, however, features nearly three times as many positive training samples with at least one figure instance annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' The remainder of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In Section 2, we first review art-historically relevant data sets that can be leveraged for image classification and object detection tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Section 3 then elaborates on the acquisition and constitution of the PoPArt data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In this context, we also clarify the annotation guidelines we adapted to the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In the course of Section 4, we address various application scenarios and discuss prospects for a digitally supported art history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Lastly, Section 5 concludes the paper and outlines areas for potential future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' The data set is available as a version-controlled repository on Zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In the field of Computer Vision, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=', 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' However, due to those collections’ highly original visuals, domain-specific, sufficiently large data sets are still required for training and fine-tuning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' The latter rely on the iconographic classification system Iconclass, which is conceived for the Western motifs of the visual arts [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' When provided, they are usually marked with bounding boxes, so that object instances are enclosed with rectangles and thus precisely located in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' To obtain more accurate information about the position of articulation points, three annotation practices have been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Reshetnikov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' [64] roughly classify poses into 12 categories, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=', by labeling human figures as sitting or kneeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Carneiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' [15], on the other hand, place additional bounding boxes around the torso and head to approximate the specifics of the human body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Only Impett and Süsstrunk [34] and Madhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' [48], however, apply fine-grained labels to faithfully represent bodily specifics by assigning keypoints on areas relevant to the figure’s pose, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=', the hips, knees, or ears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In doing so, they adhere to labeling techniques common for real-world human pose 2https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='7516230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 3For institutions from the GLAM (Galleries, Libraries, Archives, and Museums) sector that have published open access data, see the following survey: https://docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='com/spreadsheets/d/1WPS-KJptUJ-o8SXtg00llcxq0IKJu8eO6Ege_GrLaNc/edit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 4 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Schneider and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Vollmer Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Art-historically relevant data sets for image classification and object detection tasks are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Name Author(s) Year Annotation Levels Availability Formal Content Image Object Public Privat Medieval Manuscripts [89] Yarlagadda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2010 ✓ ✓ ✓ ✓1 ✓ WikiArt (f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' WikiPaintings) [85] Unknown 2010 ✓ ✓ ✓ PrintART [15] Carneiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2012 ✓ ✓ ✓1 ✓ Paintings [18] Crowley and Zisserman 2014 ✓ ✓ ✓ ✓ Picasso [28] Ginosar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2014 ✓ ✓ ✓ ✓ Painting-91 [41] Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2014 ✓ ✓ ✓ Rijksmuseum Challenge [52] Mensink and van Gemert 2014 ✓ ✓ ✓ Pandora [24] Florea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2016 ✓ ✓ ✓ Warburg’s Bilderatlas [34] Impett and Süsstrunk 2016 ✓ ✓ ✓ ✓1,2 ✓ Painter by Numbers [55] Nichol 2016 ✓ ✓ ✓ Visual Link [69] Seguin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2016 ✓ ✓ ✓ People-Art [84] Westlake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2016 ✓ ✓ ✓ ✓1 ✓ Art500k [50] Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2017 ✓ ✓ ✓ BibleVSA [3] Baraldi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2018 ✓ ✓ ✓ ✓1 ✓ ARTigo [4] Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2018 ✓ ✓ ✓ ✓ SemArt [26] Garcia and Vogiatzis 2018 ✓ ✓ ✓ ✓ IconArt [29] Gonthier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2018 ✓ ✓ ✓ ✓1 ✓ OmniArt [76] Strezoski and Worring 2018 ✓ ✓ ✓ MultitaskPainting100k [5] Bianco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2019 ✓ ✓ ✓ Ancient Chinese Art [71] Sheng and Moens 2019 ✓ ✓ ✓ ✓ Ancient Egyptian Art [71] Sheng and Moens 2019 ✓ ✓ ✓ ✓ Artpedia [75] Stefanini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2019 ✓ ✓ ✓ ✓ Iconclass Caption [16] Cetinic 2021 ✓ ✓ ✓ AQUA [27] Garcia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2020 ✓ ✓ ✓ ✓ ClassArch [48] Madhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2020 ✓ ✓ ✓ ✓1,2 ✓ Iconclass AI Test Set [59] Posthumus 2020 ✓ ✓ ✓ Saints [66] Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2020 ✓ ✓ ✓ ✓ ArtDL [53] Milani and Fraternali 2021 ✓ ✓ ✓ The Met [90] Ypsilantis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2021 ✓ ✓ ✓ ArtBench-10 [46] Liao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2022 ✓ ✓ ✓ DEArt [64] Reshetnikov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2022 ✓ ✓ ✓ ✓1 ✓ PoPArt Schneider and Vollmer 2023 ✓ ✓ ✓ ✓1,2 ✓ 1Object-level annotations include bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2Object-level annotations include keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' The Microsoft Common Objects in Context (COCO) format guidelines, for instance, require that 17 keypoints be stored with their 𝑥𝑦-coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Impett and Süsstrunk [34] extracted panels from Warburg’s Bilderatlas Mnemosyne, whereas Madhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' [48] focused on ancient Greek vase paintings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='org/#format-data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Poses of People in Art 5 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Figure detection results are reported for the People-Art test set [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' For training and validation, People-Art is used as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In contrast to previous benchmarks by Kadish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' [36] and Gonthier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' [30], we include difficult-to-annotate figures.' metadata={'source': 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0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='416 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='290 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='445 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='569 impressionistic to neo-figurative and realistic depiction styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Since our data set follows the Microsoft COCO format [47], in addition to bounding boxes, up to 17 keypoints are stored per figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Five keypoints are provided for the head, indicating the nose, eyes, and ears;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' six for the upper body, indicating wrists, elbows, and shoulders;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' and another six for the lower body, indicating ankles, knees, and hips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 3 DATA SET This section elaborates on the acquisition and constitution of the PoPArt data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' First, we outline the image collection (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='1) and annotation procedures (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' We then provide an in-depth statistical analysis of the data set (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='3) and present its underlying data format (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='1 Image Collection Like many authors before, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=', Westlake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' [84] and Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' [50], we exploit the art-historical online encyclopedia WikiArt [85] as content provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' (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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' (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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In a preliminary step, we extracted images from WikiArt that have a high probability of depicting human figures, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=', images on which at least one figure can be automatically detected with a probability of 𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' To this end, we benchmarked the suitability of models commonly used for object detection and applied the best-performing one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' For evaluation, we use the metrics and tools provided by the COCO API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='5 All models were first pre-trained on the Microsoft COCO 2017 data set6 for 12 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' As optimization algorithms, we employed Stochastic Gradient Descent (SGD) for ResNet-50 and Adam [42] for Transformer backbones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' momentum and weight decay were set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='9 and 1e − 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' The initial 5https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='com/cocodataset/cocoapi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 6https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='com/datasets/awsaf49/coco-2017-dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 6 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Schneider and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' The PoPArt data set contains 22 depiction styles, ranging from impressionistic to neo-figurative and realistic variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' For each style, an exemplary image is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' All images originate from the art-historical online encyclopedia WikiArt [85] and are in the public domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Poses of People in Art 7 (a) Data set view (b) Annotation view Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' learning rate decays at the 8th and 11th epoch with 2e − 2 set for ResNet-50-backed and 1e − 4 for Transformer- backed architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Models were then fine-tuned, with their classification head re-initialized, for another 12 epochs on People-Art [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' The learning rate is decreased to 2e − 4 in case of ResNet-50 and 1e − 5 in case of Transformer backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' (ii) either RGBShift or HueSaturationValue is applied with 𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' (iii) JpegCompression is applied with 𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' (iv) ChannelShuffle is applied with 𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' and (v) either Blur or MedianBlur is applied with 𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Images are reduced to a maximum scale of 1, 333 × 800 pixels without changing the aspect ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In contrast to previous studies by Kadish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' [36] and Gonthier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' [30], we include difficult-to-annotate figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='7 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='8 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' At a more restrictive Intersection over Union (IoU) threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='75, the difference increases further, rising to between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='4 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' This effect also is noticeable with Average Recall (AR), which is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='5 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='8 % higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Since TOOD surpasses PVT in AR by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='4 %, with AP being almost equal, we assume that it is generally suited best to the stylistic peculiarities of the art-historical domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' A maximum of 125 images per style were then selected for image annotation, taking into account the sampling distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Exact-duplicate and near-duplicate reproductions were removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' For each style, an example image is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Following Everingham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' [21], we designed the annotation procedure to be as (i) exhaustive, (ii) consistent, and (iii) accurate as possible, without omitting art-historical depiction specifics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' O edward-hopper_new-york-restaurant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='jpg kuzma-petrov-vodkin_costume-design-for-the-tragedy-of-pu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='. O marc-chagal_adam-and-eve-with-the-forbidden-fruit-1960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='. 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='jpg O titian_virgin-and-child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='jpg O paolo-veronese_venus-and-adonis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='jpg 1 annotation (8 keypoints 2 annotations（26 keypoints 3 annotations 46 keypoints 1 annotation )（13 keypoints (159 1160 (1161 O theophrastos-triantafyllidis_friends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='jpg O zinaida-serebriakova_portrait-of-aleksandr-serebriakov-stud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' : O vladimir-borovikovsky_portrait-of-a-and-v-gagarin-1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='jpg O maurice-de-vlaminck_portrait-of-a-woman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='jpg 5 annotations（15 keypoints 1 annotation（9 keypoints 2 annotations（26 keypoints 1 annotation（7 keypoints 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='.121314 (15 16 17) 480COCO ANNOTATOR DATASETSPOPARTCATEGORIES U SSCHNEIDER 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' tintoret_the-birth-fjohn-the-baptist jpg QPERSON(11) tintoretto_the-birth-of-john-the-baptist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Schneider and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Vollmer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Challenges ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Variations of the size ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='of human figures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Large crowds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Small figures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Figures hard to separate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='from each other ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Figures difficult to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='recognize as such ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Image-extrinsic factors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Image-intrinsic factors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Figures in the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='background ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Relation of human ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='figures to each other ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Referencing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Shadows ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Reflections ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='In water ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='In the mirror ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='On other surfaces ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='(such as helmets) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Not referencing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Overlaps ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Intersections ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Symmetrically ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='arranged figures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Deviations from the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='‘ideal’ human body ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Stylistic variance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Lack of differentiation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='of the face ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Lack of differentiation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='of the body shape ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Veiled body ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Non-human bodies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='and body parts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Human-like animals ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Mythological figures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Biblical figures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Non-living bodies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='and body parts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Fabricated bodies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Dolls ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Masks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Crafts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Sculptures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Skeletons ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Severed heads ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Severed limbs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Image-extrinsic factors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Image-intrinsic factors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Positioning of the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='human body ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Back views ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Profile views ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Distortions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Twists and turns ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Annotator [7], we used a web-based open-source tool for bounding box and keypoint annotation that we minimally adapted to our needs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='1 Exhaustiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' We set the following guidelines to guarantee exhaustive annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' (i) All human- appearing figures are enclosed by bounding boxes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' the distance to the outline of the human figure is to be kept as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Only the visible area of the figure is labeled and not the estimated total extent of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' All images are in the public domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' do not indicate truncated or difficult-to-annotate figures as such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' (ii) Up to four human figures per image are fine-granularly labeled with keypoints, selecting those whose limbs can be captured best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Keypoints are recorded in a ‘person-centric’ way, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=', left points refer to the figure’s left extremities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Since in many cases keypoints are not clearly visible or are occluded, we establish three rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' (a) If an occluded body part can be approximated by another, it is denoted by a keypoint;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=', an elbow obscured by a pillar is annotated if the hand and shoulder of the respective body half are visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' (c) If several joints are not visible and cannot be approximated, the corresponding keypoints are not set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='2 Consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' To ensure consistency in the annotation, a fixed team of annotators was employed at the Ludwig Maximilian University of Munich throughout the entire period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Annotation guidelines were discussed with the annotators prior to annotation and iteratively modified during the annotation procedure, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=', when unusual figure constellations occurred more frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In the course of the process, recurring challenges arose for both modes, bounding box and keypoint annotation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 4 visualizes them in taxonomic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Schneider and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Variations of the size of human figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Large crowds and figures in the background complicate the annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Both cases are dominated by very small figures (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 5a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 5b), figures that are difficult to separate from each other (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 5c), or that are difficult to recognize as human (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 5d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' The latter is due not only to factors intrinsic to the object, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=', the analog original, but also to image-extrinsic factors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=', the original’s digital reproduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In particular, compression artifacts or low-quality and out-of-date resolutions hamper the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Relation of human figures to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' We distinguish two kinds of figure relations, which are crucial for annotation: non-referential and referential ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Referential relations include constellations in which the body of one and the same figure is represented several times but in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In addition to shadows (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 5e), these mainly include reflections, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=', in mirrors (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 5f), in water (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 5g), and on surfaces like metallic armor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' We set the corresponding bounding boxes whenever the referencing part, the reflection, can be recognized as human-like even without the referenced part, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=', the human reflected in some way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Non-referential relations are found when figures overlap, intersect, or are symmetrically arranged (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 5h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In case of overlaps and intersections, we approximate occluded keypoints as far as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Deviations from the ‘ideal’ human body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' We declare bodies as deviating from this ideal whose depicted measurements or proportions do not adhere to usual conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Three subcategories are discerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' (a) Often deviations are due to stylistic reasons expressed regionally, epochally, or individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' The lack of differentiation of the entire body shape or individual body parts is characteristic of Impressionism and Pointillism (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 5i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' If the placement of keypoints in an image is complicated by blurred contours, distorted proportions, or missing joints, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 5j, we approximate them, provided the figures can be recognized as human.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' (b) Another subcategory comprises non-human bodies and body parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' These include mythological figures such as centaurs, harpies, and mermaids (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 5k), biblical figures, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=', angels, and human-like animals like monkeys and lemurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' While animals are excluded from annotation, we annotate human parts of mythological and biblical figures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' consequently, the animal limbs of centaurs are not annotated, nor are the wings and halo of angels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' (c) The third subcategory covers non-living bodies and body parts, with a considerable portion being severed heads (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 5l)7 and limbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=', in particularly detailed views or images in need of restoration that no longer permit keypoints to be fully labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Also included are fabricated bodies, such as dolls, masks, crafts, sculptures, and images within images depicting human bodies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=', in salon paintings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Positioning of the human body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Of relevance is the body’s positioning in the image space especially for back views, as in Dürer’s Hercules (1498;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 5m), where the inversion of keypoints must be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' For profile views, it is crucial to set the eyes and ears on the non-visible side of the face as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' This applies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=', to Florentine portraits (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 5n), which refer to the strict profile of emperors on ancient coins [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In a considerable number of images, perspective distortions are furthermore present, along with twists and turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' They are found primarily in Baroque and Rococo works such as those by Tiepolo (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 5o), but also in 20th-century avant-garde 7See iconographies such as David and Goliath, Judith and Holofernes, and Salomé and John the Baptist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Poses of People in Art 11 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' The People-Art [84] and PoPArt data sets are descriptively compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Figures are indicated by bounding boxes associated with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Up to 17 keypoints are stored per figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Difficult-to-annotate figures are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Data set Split Images Images𝑃𝑜𝑠 Images𝑁𝑒𝑔 Figures Crowds Keypoints Styles People-Art Training 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='623 525 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='098 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='512 0 0 43 Validation 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='383 442 941 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='219 0 0 43 Testing 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='616 522 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='094 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='137 0 0 43 Total 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='622 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='489 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='133 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='868 0 0 43 PoPArt Training 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='472 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='472 0 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='457 245 33,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='582 22 Validation 491 491 0 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='175 114 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='104 22 Testing 491 491 0 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='117 106 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='468 22 Total 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='454 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='454 0 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='749 465 56,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='154 22 movements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' as in the French surrealist André Masson (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 5p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' There, too, the possible inversion of keypoints has to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='3 Accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Figure instance annotations were checked in several test cycles according to formerly stated guidelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' They were once again reviewed at the end of the annotation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' We verified, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=', 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='3 Descriptive Statistics For machine learning purposes, the PoPArt data set is divided into three subsets: training, validation, and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' They contain 1,472, 491, and 491 images, respectively, so approximate split ratios of 60 %, 20 %, and 20 % are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' The widest image measures 6,298 × 3,049 and the highest 4,524 × 6,018 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Table 3 summarizes PoPArt in comparison to the similarly constituted People-Art data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Both data sets focus exclusively on human figures depicted in art-historical objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' other classes are not annotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Further structural differences arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' (i) People-Art contains more training, validation, and testing images due to the integration of negative image samples that do not show human figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' However, PoPArt features almost three times as many positive samples in the training set, which have at least one instance annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' This includes more small-area instances measuring between 0 and 162 pixels, namely 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='9 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In the People-Art data set, it amounts to only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='5 % despite reduced image sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In addition, the data set completely lacks crowd annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' PoPArt thus decisively enables the automatic detection even of figures that are displayed small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' (ii) Moreover, PoPArt accounts for the broad spectrum of art-historical body language in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 6, the proportion of images with at least six figures is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='52 % higher in 12 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Schneider and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' The proportion of images with at least six figures is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='52 % higher in the PoPArt than in the People-Art data set [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' This is also reflected in a larger maximum number of figures in an image: it is 28 for People-Art and 110 for PoPArt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' The PoPArt data set has a 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' the PoPArt than in the People-Art data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' This is reflected in a larger maximum number of figures in an image: it is 28 for People-Art and 110 for PoPArt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' As a result, the PoPArt data set also has a 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='22 % higher share of overlapping figure instances than People-Art (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 7), with the maximum number of overlaps in an image being 11 for People-Art and 23 for PoPArt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='4 Data Split Format All data splits follow the JSON-based Microsoft COCO format [47];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 8 displays an exemplary figure instance annotation with its referencing image annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Figure instance annotations contain task-agnostic information (id, image_id, and category_id), supplemented by fields essential to the respective detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In contrast to the Microsoft COCO format guidelines, we assign 𝑣 = 2 to index occluded keypoints as well, rather than 𝑣 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' jpg", "metadata": { "wikiart_url": "https ://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='wikiart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='org/en/albrecht -durer/death -of -orpheus -1498" , "wikiart_image_url": "https :// uploads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='wikiart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='org/images/albrecht -durer/death -of -orpheus 1498.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' jpg", "wikiart_style": "Northern Renaissance" } }, · · ], "annotations": [ · · { "id": 64, "image_id": 58, "category_id": 1, "area": 1319355, "bbox": [ 616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='0, 1718.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='0, 1305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='0, 1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='0 ], "segmentation": [ [ 1921.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='0, 1718.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='0, 1921.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='0, 2729.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='0, 616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='0, 2729.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='0, 616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='0, 1718.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' PoPArt follows the JSON-based Microsoft COCO format [47], for which a figure instance annotation with its referencing image annotation is displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Two scenarios are distinguished: those arising from human pose estimation (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='1), and those from human figure detection (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='1 Human Pose Estimation In a first application scenario, we demonstrate that PoPArt enables the quantitatively systematized exploration of human poses in visual art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' For this purpose, we suggested in Springstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' [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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' We in this context adapted a semi-supervised 14 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Schneider and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' The two-stage human pose estimator from Springstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' [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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' The individual boxes are cropped and serve as input for the second stage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' the second Transformer model then computes a set of keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' (a) Default image grid (b) Two-dimensional canvas view Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' The basic principle is to use both labeled and unlabeled image material to train a student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' The teacher serves as a generator of pseudo-labels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Moreover, in a user study, we also confirmed the feasibility of the approach for retrieval tasks, enabling the search for resembling poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Figure detection results are reported for the People-Art test set [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' For training and validation, PoPArt was used in addition to People-Art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In contrast to previous benchmarks by Kadish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' [36] and Gonthier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' [30], we include difficult-to-annotate figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' The best performing approach is indicated in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Model Backbone LR AP AP50 AP75 AP𝑆 AP𝑀 AP𝐿 AR TOOD [22] ResNet-50-FPN 2e − 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='478 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='780 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='499 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' This becomes particularly evident when machine-generated similarity arrangements are explored through web-based user interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' First, the retrieval of certain iconographies is thereby enabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Several cluster structures emerge in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 10b: the one shown at the top left, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=', reveals an image group of more dynamic poses that are conspicuous for their bent or flared legs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' apart from the fact that here the apple is being handed to Adam in a rather prominent manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' For this purpose, we utilize the same models and pipeline as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Parameter settings remain unchanged;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' the learning rate is, again, set to 2e − 4 in case of ResNet-50 and 1e − 5 in case of Transformer backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' For the Transformer-based PVT model [81], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=', AP and AR improve to the same extent, from 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='5 to 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='7 % and 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='1 to 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='5 %, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' The leap is even more noticeable if we plug-in the PoPArt instead of the People-Art test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' AP then rises from 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='6 to 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='6 % and AR from 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='2 to 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='0 % for PVT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' The additional integration 8https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='iart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='vision/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 16 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Schneider and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Vollmer (a) Nicholas Poussin, The Deluge (1660–1664) (b) John Martin, The Deluge (1834) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Detail views of the crowds depicted in Nicholas Poussin’s and John Martin’s versions of The Deluge, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Both images have been slightly lightened to emphasize depiction specifics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' The images are in the public domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' in Northern Renaissance works, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=', AP improves from 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='1 to 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='5 % and AR from 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='5 to 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='4 % (Table 5 in Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' even if these may initially only be used to pre-filter the (digitally available) image material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' John Martin’s emphatically apocalyptic Deluge (1834;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In Nicholas Poussin’s Deluge (1660–1664;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 11a), on the other hand, the majority of figures are still, because of the larger body size, differentiated in their moments of action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' While a man clings to his horse in the foreground, a mother, slightly moved back, stretches her child upwards to the shore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Recommender systems like iART [67, 74] can ultimately point to research-worthy phenomena here as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In addition to mandatory fields, image annotations provide metadata from the art-historical online encyclopedia WikiArt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was funded in part by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) under project no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 415796915.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' We thank Ursula Huber for her valuable support with the image annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' We also thank Hubertus Kohle, Ralph Ewerth, and Matthias Springstein for fruitful discussions and useful comments on the subject matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' AUTHORS’ CONTRIBUTIONS S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' conceived, designed, and performed the experiments, analyzed the data, oversaw image annotation, and ensured data quality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' performed image annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' and R.' metadata={'source': 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+page_content=' IEEE, New York, 11580–11588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='1109/CVPR42600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='01160 [14] Nicolas Carion, Francisco Massa, Gabriel Synnaeve, Nicolas Usunier, Alexander Kirillov, and Sergey Zagoruyko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' End-to-end Object Detection with Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In Computer Vision – ECCV 2020 (Lecture Notes in Computer Science, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 12346), Andrea Vedaldi, Horst Bischof, Thomas Brox, and Jan-Michael Frahm (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Springer, Cham, 213–229.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Springer, Cham, 143–157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='1007/978-3-642-33765-9_11 [16] Eva Cetinic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Iconographic Image Captioning for Artworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In Pattern Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' ICPR International Workshops and Challenges (Lecture Notes in Computer Science, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 12663), Alberto Del Bimbo, Rita Cucchiara, Stan Sclaroff, Giovanni Maria Farinella, Tao Mei, Marco Bertini, Hugo Jair Escalante, and Roberto Vezzani (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Springer, Cham, 502–516.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='1007/978-3-030-68796-0_36 [17] Keith Christiansen and Patricia Lee Rubin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Gesichter der Renaissance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Meisterwerke italienischer Portrait-Kunst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Published following the exhibition “Gesichter der Renaissance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Meisterwerke italienischer Portrait-Kunst” at the Bode Museum Berlin, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='– 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='2011 and at the Metropolitan Museum of Art New York, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='2011–18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' [18] Elliot J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Crowley and Andrew Zisserman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In Search of Art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In Computer Vision – ECCV 2014 Workshops (Lecture Notes in Computer Science, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 8925), Lourdes Agapito, Michael M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Bronstein, and Carsten Rother (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Springer, Cham, 54–70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='1007/978- 3-319-16178-5_4 [19] Charles Darwin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 1877.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Der Ausdruck der Gemüthsbewegungen bei dem Menschen und den Thieren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Schweizerbart, Stuttgart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Retrieved January 13, 2023 from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='zum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='de/stueber/darwin/werke07/icon_page_00000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='html [20] James Elkins (Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2007.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='1007/s11263-009-0275-4 [22] Chengjian Feng, Yujie Zhong, Yu Gao, Matthew R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Scott, and Weilin Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' TOOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Task-aligned One-stage Object Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In IEEE/CVF International Conference on Computer Vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' IEEE, New York, 3510–3519.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='1109/ICCV48922.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='00349 [23] Ilsebill Barta Fliedl and Christoph Geissmar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Die Beredsamkeit des Leibes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Zur Körpersprache in der Kunst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Residenz Verlag, Salzburg/Wien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' [24] Corneliu Florea, Razvan George Condorovici, Constantin Vertan, Raluca Butnaru, Laura Florea, and Ruxandra Vrânceanu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Pandora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Description of a Painting Database for Art Movement Recognition with Baselines and Perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In 24th European Signal Processing Conference, EUSIPCO 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' IEEE, New York, 918–922.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='1109/EUSIPCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='7760382 [25] Marcus Frings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Mensch und Maß.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Anthropomorphe Elemente in der Architekturtheorie des Quattrocento.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Verlag und Datenbank für Geisteswissenschaften, Weimar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' [26] Noa Garcia and George Vogiatzis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' How to Read Paintings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Semantic Art Understanding with Multi-modal Retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In Computer Vision – ECCV 2018 Workshops (Lecture Notes in Computer Science, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 11130), Laura Leal-Taixé and Stefan Roth (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Springer, Cham, 676–691.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='1007/978-3-030-11012-3_52 [27] Noa Garcia, Chentao Ye, Zihua Liu, Qingtao Hu, Mayu Otani, Chenhui Chu, Yuta Nakashima, and Teruko Mitamura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' A Dataset and Baselines for Visual Question Answering on Art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In Computer Vision – ECCV 2020 Workshops (Lecture Notes in Computer Science, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 12536), Adrien Bartoli and Andrea Fusiello (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Springer, Cham, 92–108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='1007/978-3-030-66096-3_8 [28] Shiry Ginosar, Daniel Haas, Timothy Brown, and Jitendra Malik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2014.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='1007/978-3-319-16178-5_7 [29] Nicolas Gonthier, Yann Gousseau, Saïd Ladjal, and Olivier Bonfait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Weakly Supervised Object Detection in Artworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In Computer Vision – ECCV 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Kritische Gesamtausgabe des Bauhüttenbuches ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' fr 19093 der Pariser Nationalbibliothek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Akademische Druck- und Verlagsanstalt, Graz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' [32] Berthold Hinz (Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2011.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='edu/ LiteraryLabPamphlet16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='pdf [34] Leonardo Impett and Sabine Süsstrunk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Pose and Pathosformel in Aby Warburg’s Bilderatlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In Computer Vision – ECCV 2016 Workshops (Lecture Notes in Computer Science, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='1109/TPAMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2577031 [64] Artem Reshetnikov, Maria-Cristina Marinescu, and Joaquim More Lopez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' DEArt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Dataset of European Art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='01226 [65] Hans Rupprich (Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 1966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Die Anfänge der theoretischen Studien / Das Lehrbuch der Malerei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Von der Maß der Menschen, der Pferde, der Gebäude;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Von der Perspektive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Von Farben / Ein Unterricht alle Maß zu ändern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Dürer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Schriftlicher Nachlass, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Deutscher Verein für Kunstwissenschaft, Berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' [66] Stefanie Schneider, Matthias Springstein, Javad Rahnama, Eyke Hüllermeier, Ralph Ewerth, and Hubertus Kohle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' The Dissimilar in the Similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' An Attribute-guided Approach to the Subject-specific Classification of Art-historical Objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Jahrestagung der Gesellschaft für Informatik, INFORMATIK 2020 (LNI, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' P-307), Ralf H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Reussner, Anne Koziolek, and Robert Heinrich (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' GI, Bonn, 1355–1364.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='18420/inf2020_127 [67] Stefanie Schneider, Matthias Springstein, Javad Rahnama, Hubertus Kohle, Ralph Ewerth, and Eyke Hüllermeier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' iART.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Eine Such- maschine zur Unterstützung von bildorientierten Forschungsprozessen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Tagung des Verbands Digital Humanities im deutschsprachigen Raum, Michaela Geierhos (Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 142–147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='6304590 [68] Erhard Schön.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 1920.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Unterweisung der Proportion und Stellung der Possen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Frankfurt am Main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' [69] Benoit Seguin, Carlotta Striolo, Isabella diLenardo, and Frédéric Kaplan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Visual Link Retrieval in a Database of Paintings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In Computer Vision – ECCV 2016 Workshops (Lecture Notes in Computer Science, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 9913), Gang Hua and Hervé Jégou (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Springer, Cham, 753–767.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='1007/978-3-319-46604-0_52 [70] Xi Shen, Alexei A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Efros, and Mathieu Aubry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Discovering Visual Patterns in Art Collections With Spatially-consistent Feature Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In IEEE Conference on Computer Vision and Pattern Recognition, CVPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' IEEE, New York, 9278–9287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='1109/ CVPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='00950 [71] Shurong Sheng and Marie-Francine Moens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Generating Captions for Images of Ancient Artworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In MM ’19: The 27th ACM International Conference on Multimedia, Laurent Amsaleg, Benoit Huet, Martha A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Larson, Guillaume Gravier, Hayley Hung, Chong-Wah Ngo, and Wei Tsang Ooi (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' ACM, New York, 2478–2486.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='1145/3343031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='3350972 [72] Klaus Speich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 1957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Die Proportionslehre des menschlichen Körpers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Antike, Mittelalter, Renaissance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Dissertation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Universität Zürich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' [73] Matthias Springstein, Stefanie Schneider, Christian Althaus, and Ralph Ewerth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Semi-supervised Human Pose Estimation in Art-historical Images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In MM ’22: The 30th ACM International Conference on Multimedia, João Magalhães, Alberto Del Bimbo, Shin’ichi Satoh, Nicu Sebe, Xavier Alameda-Pineda, Qin Jin, Vincent Oria, and Laura Toni (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' ACM, New York, 1107–1116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='1145/3503161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='3548371 [74] Matthias Springstein, Stefanie Schneider, Javad Rahnama, Eyke Hüllermeier, Hubertus Kohle, and Ralph Ewerth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' iART.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' A Search Engine for Art-Historical Images to Support Research in the Humanities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In MM ’21: The 29th ACM International Conference on Multimedia, Heng Tao Shen, Yueting Zhuang, John R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Smith, Yang Yang, Pablo César, Florian Metze, and Balakrishnan Prabhakaran (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' ACM, New York, 2801–2803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='1145/3474085.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 11752), Elisa Ricci, Samuel Rota Bulò, Cees Snoek, Oswald Lanz, Stefano Messelodi, and Nicu Sebe (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Springer, Cham, 729–740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='1007/978-3-030-30645-8_66 [76] Gjorgji Strezoski 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Vanschoren and Sai-Kit Yeung (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 12 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Retrieved January 13, 2023 from https://datasets-benchmarks-proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='neurips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='cc/paper/2021/file/5f93f983524def3dca464469d2cf9f3e-Paper-round2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content='pdf [91] Linda Gertner Zatlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Beardsley, Japonisme and the Perversion of the Victorian Ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Cambridge University Press, Cambridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' [92] Frank Zöllner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Anthropomorphismus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Das Maß des Menschen in der Architektur von Vitruv bis Le Corbusier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In Ist der Mensch das Maß aller Dinge?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Beiträge zur Aktualität des Protagoras, Otto Neumaier (Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Bibliopolis, Möhnesee, 306–344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' APPENDIX Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' Figure detection results are reported for the PoPArt test set by depiction style and training set(s);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' TOOD [22] is employed as model, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' In contrast to previous benchmarks by Kadish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' [36] and Gonthier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf'} +page_content=' [30], we include difficult-to-annotate figures.' metadata={'source': 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0000000000000000000000000000000000000000..4fed875061916882e85f1753a82306ae2e3e1bfc --- /dev/null +++ b/INFJT4oBgHgl3EQfFiwP/content/tmp_files/2301.11443v1.pdf.txt @@ -0,0 +1,7023 @@ +LIMITLESS STABILITY FOR GRAPH CONVOLUTIONAL +NETWORKS +Christian Koke +Technical University of Munich +christian.koke@tum.de +ABSTRACT +This work establishes rigorous, novel and widely applicable stability guarantees +and transferability bounds for graph convolutional networks – without reference +to any underlying limit object or statistical distribution. Crucially, utilized graph- +shift operators (GSOs) are not necessarily assumed to be normal, allowing for the +treatment of networks on both directed- and for the first time also undirected graphs. +Stability to node-level perturbations is related to an ’adequate (spectral) covering’ +property of the filters in each layer. Stability to edge-level perturbations is related to +Lipschitz constants and newly introduced semi-norms of filters. Results on stability +to topological perturbations are obtained through recently developed mathematical- +physics based tools. As an important and novel example, it is showcased that +graph convolutional networks are stable under graph-coarse-graining procedures +(replacing strongly-connected sub-graphs by single nodes) precisely if the GSO is +the graph Laplacian and filters are regular at infinity. These new theoretical results +are supported by corresponding numerical investigations. +1 +INTRODUCTION +Graph Convolutional Networks (GCNs) (Kipf & Welling, 2017; Hammond et al., 2011; Defferrard +et al., 2016) generalize Euclidean convolutional networks to the graph setting by replacing con- +volutional filters by functional calculus filters; i.e. scalar functions applied to a suitably chosen +graph-shift-oprator capturing the geometry of the underlying graph. A key concept in trying to +understand the underlying reasons for the superior numerical performance of such networks on graph +learning tasks (as well as a guiding principle for the design of new architectures) is the concept +of stability. In the Euclidean setting, investigating stability essentially amounts to exploring the +variation of the output of a network under non-trivial changes of its input (Mallat, 2012; Wiatowski +& Bölcskei, 2018). In the graph-setting, additional complications are introduced: Not only input +signals, but now also the graph shift operators facilitating the convolutions on the graphs may vary. +Even worse, there might also occur changes in the topology or vertex sets of the investigated graphs +– e.g. when two dissimilar graphs describe the same underlying phenomenon – under which graph +convolutional networks should also remain stable. This last stability property is often also referred +to as transferability (Levie et al., 2019a). Previous works investigated stability under changes in +graph-shift operators for specific filters (Levie et al., 2019b; Gama et al., 2020) or the effect of +graph-rewiring when choosing a specific graph shift operator (Kenlay et al., 2021). Stability to +topological perturbations has been established for (large) graphs discretising the same underlying +topological space (Levie et al., 2019a), the same graphon (Ruiz et al., 2020; Maskey et al., 2021) or +for graphs drawn from the same statistical distribution (Keriven et al., 2020; Gao et al., 2021). +Common among all these previous works are two themes limiting practical applicability: First and +foremost, the class of filters to which results are applicable is often severely restricted. The same is +true for the class of considered graph shift operators; with non-normal operators (describing directed +graphs) either explicitly or implicitly excluded. Furthermore – when investigating transferability +properties – results are almost exclusively available under the assumption that graphs are large and +either discretize the same underlying ’continuous’ limit object suffieciently well, or are drawn from +the same statistical distributions. While these are of course relevant regimes, they do not allow to +draw conclusions beyond such asymptotic settings, and are for example unable to deal with certain +spatial graphs, inapplicable to small-to-medium sized social networks and incapable of capturing +1 +arXiv:2301.11443v1 [cs.LG] 26 Jan 2023 + +the inherent multi-scale nature of molecular graphs (as further discussed below). Finally, hardly any +work has been done on relating the stability to input-signal perturbations to network properties such +as the interplay of utilized filters or employed non-linearities. The main focus of this work is to +provide alleviation in this situation and develop a ’general theory of stability’ for GCNs – agnostic +to the types of utilized filters, graph shift operators and non-linearities; with practically relevant +transferability guarantees not contingent on potentially underlying limit objects. To this end, Section +2 recapitulates the fundamentals of GCNs in a language adapted to our endeavour. Sections 3 and +4 discuss stability to node- and edge-level perturbations. Section 5 discusses stability to structural +perturbations. Section 6 discusses feature aggregation and Section 7 provides numerical evidence. +2 +GCNS VIA COMPLEX ANALYSIS AND OPERATOR THEORY +Throughout this work, we will use the label G to denote both a graph and its associated vertex set. +Taking a signal processing approach, we consider signals on graphs as opposed to graph embeddings: +Node-Signals: +Node-signals on a graph are then functions from G to the complex numbers; i.e. +elements of C|G| (with |G| the cardinality of G). We allow nodes i P G in a given graph to have +weights µi not necessarily equal to one and equip the space C|G| with an inner product according to +xf, gy “ ř +iPG fpiqgpiqµi to account for this. We denote the hence created Hilbert space by ℓ2pGq. +Characteristic Operators: +Fixing an indexing of the vertices, information about connectivity +within the graph is encapsulated into the set of edge weights, collected into the adjacency matrix W +and (diagonal) degree matrix D. Together with the weight matrix M :“ diag +´ +tµiu|G| +i“1 +¯ +, various +standard geometry capturing characteristic operators – such as weighted adjacency matrix M ´1W, +graph Laplacian ∆ :“ M ´1pD ´ Wq and normalized graph Laplacian L :“ M ´1D´ 1 +2 pD ´ +WqD´ 1 +2 can then be constructed. For undirected graphs, all of these operators are self-adjoint. On +directed graphs, they need not even be normal (T ˚T “ TT ˚). We shall remain agnostic to the choice +of characteristic operator; differentiating only between normal and general operators in our results. +Functional Calculus Filters: +A crucial component of GCNs are functional calculus filters, which +arise from applying a function g to an underlying characteristic operator T; creating a new operator +gpTq. Various methods of implementations exist, all of which agree if multiple are applicable: +GENERIC FILTERS: +If (and only if) T is normal, we may apply generic complex valued functions +g to T: Writing normalized eigenvalue-eigenvector pairs of T as pλi, φiq|G| +i“1 one defines gpTqψ “ +ř|G| +i“1 gpλiqxφi, ψyℓ2pGqφi for any ψ P ℓ2pGq. One has }gpTq}op “ supλPσpT q |gpλq|, with σpTq +denoting the spectrum of T. If g is bounded, one may obtain the T-independent bound }gpTq}op ď +}g}8. Keeping in mind that g being defined on all of σpTq (as opposed to all of C) is clearly sufficient, +we define a space of filters which will harmonize well with our concept of transferability discussed in +Section 5. The introduced semi-norm will quantify the stability to perturbations in coming sections. +Definition 2.1. Fix ω P C and C ą 0. Define the space F cont +ω,C of continuous filters on Cztω, ωu, +to be the space of multilinear power-series’ gpzq “ ř8 +µ,ν“0 aµν pω ´ zq´µ pω ´ zq´µ for which the +semi-norm }g}F cont +ω,C :“ ř8 +µ,νą0 |µ ` ν|Cµ`ν´1|aµν| is finite. +Denoting by Bϵpωq Ď C the open ball of radius ϵ around ω, one can show that for arbitrary δ ą 0 and +every continuous function g defined on CzpBϵpωq Y Bϵpωqq which is regular at infinity – i.e. satisfies +limrÑ`8 gprzq “ c P C independent of which z ‰ 0 is chosen – there is a function f P F cont +ω,C +so that |fpzq ´ gpzq| ď δ for all z P CzpBϵpωq Y Bϵpωqq. In other words, functions in F cont +ω,C can +approximate a wide class of filters to arbitrary precision. More details are presented in Appendix B. +ENTIRE FILTERS: +If T is not necessarily normal, one might still consistently apply entire (i.e. +everywhere complex differentiable) functions to T. Detail details on the mathematical background are +given in Appendix C. Here we simply note that such a function g is representable as an (everywhere +convergent) power series gpzq :“ ř8 +k“0 ag +kzk so that we may simply set gpTq “ ř8 +k“0 ag +k ¨ T k. For +2 + +the norm of the derived operator one easily finds }gpTq}op ď ř8 +k“0 |ag +k|}T}k +op using the triangle +inequality. While entire filters have the advantage that they are easily and efficiently implementable – +making use only of matrix multiplication and addition – they suffer from the fact that it is impossible +to give a }T}op-independent bound for }gpTq}op as for continuous filters. This behaviour can be +traced back to the fact that no non-constant bounded entire function exists (Bak & Newman, 2017). +HOLOMORPHIC FILTERS: +To define functional calculus filters that are both applicable to non- +normal T and boundable somewhat more controlably in terms of T, one may relax the condition +that g be entire to demanding that g be complex differentiable (i.e. holomorphic) only on an +open subset U Ď C of the complex plane. Here we assume that U extends to infinity in each +direction (i.e. is the complement of a closed and bounded subset of C). For any g holomor- +phic on U and regular at infinity we set (with pzId ´ Tq´1 the so called reolvent of T at z) +gpTq :“ gp8q ¨ Id ` +1 +2πi +¿ +BD +gpzq ¨ pzId ´ Tq´1dz, +(1) +for any T whose spectrum σpTq is completely contained in U. Here +we have used the notation gp8q “ limrÑ`8 gprzq and taken D to an +open set with nicely behaved boundary BD (more precisely a Cauchy +domain; c.f. Appendix C). We assume that D completely contains σpTq +and that its closure D is completely contained in U. The orientation +Figure 1: Set-Visualisations +of the boundary BD is the usual positive orientation on D (such that D ’is on the left’ of BD; cf. Fig. +1). Using elementary facts from complex analysis it can be shown that the resulting operator gpTq in +(1) is independent of the specific choice of D (Gindler, 1966). While we will present results below in +terms of this general definition – remaining agnostic to numerical implementation methods for the +most part – it is instructive to consider a specific exemplary setting with definite and simple numerical +implementation of such filters: To this end, chose an arbitrary point ω P C and set U “ Cztωu in the +definitions above. Any function g that is holomorphic on U and regular at 8 may then be represented +by its Laurent series, which is of the form gpzq “ ř8 +k“0 bg +kpz ´ ωq´k (Bak & Newman, 2017). For +any T with σpTq Ď U (i.e. ω R σpTq) evaluating the integral in (1) yields (c.f. Appendix C): +gpTq “ +8 +ÿ +k“0 +bg +k ¨ pT ´ ωIdq´k +(2) +Such filters have already been employed successfully, e.g. in the guise of Cayley filters (Levie et al., +2019c), which are polynomials in z`i +z´i “ 1 ` +2i +z´i. We collect them into a designated filter space: +Definition 2.2. For a function gpzq “ ř8 +k“0 bg +kpz ´ ωq´k on U :“ Cztωu define the semi-norm +}g}F hol +ω,C :“ ř8 +k“1 |bg +k|kCk´1 for C ą 0. Denote the set of such g for which }g}F hol +ω,C ă 8 by F hol +ω,C. +In order to derive }T}op-independent bounds for }gpTq}op, we will need to norm-bound the resolvents +appearing in (1) and (2). If T is normal, we simply have }pzId ´ Tq´1}op “ 1{distpz, σpTqq. In the +general setting, following Post (2012), we call any positive function γT satisfying }pzId´Tq´1}op ď +γT pzq on CzσpTq a resolvent profile of T. Various methods (e.g. Szehr (2014); MichaelGil (2012)) +to find resolvent profiles. Most notably Bandtlow (2004b) gives a resolvent profile solely in terms of +1{distpz, σpTqq and the departure from normality of T. We then find the following result: +Lemma 2.3. For holomorphic g and generic T we have }gpTq}op ď |gp8q|` 1 +2π +ű +BD |gpzq|γT pzqd|z|. +Furthermore we have for any T with γT pωq ď C, that }gpTq}op ď }g}F hol +ω,C as long as g P F hol +ω,C. +Lemma 2.3 (proved in Appendix D) finally bounds }gpTq}op independently of T, as long as appearing +resolvents are suitably bounded; which – importantly – does not force }T}op to be bounded. +Non-Linearities & Connecting Operators: +To each layer of our GCN, we associate a (possibly) +non-linear and Ln-Lipschitz-continuous function ρn : C Ñ C satisfying ρnp0q “ 0 which acts +point-wise on signals in ℓ2pGnq. This definition allows to choose ρn “ | ¨ |, ReLu, Id or any sigmoid +function shifted to preserve zero. To account for recently proposed networks where input- and +’processing’ graphs are decoupled (Alon & Yahav, 2021; Topping et al., 2021), and graph pooling +layers (Lee et al., 2019), we also allow signal representations in the hidden network layers n to live in +3 + +D +OD +(T) +au +U\D +g(T)varying graph signal spaces ℓ2pGnq. Connecting operators are then (not necessarily linear) operators +Pn : ℓ2pGn´1q Ñ ℓ2pGnq connecting the signal utilized of subsequent layers. We assume them to be +Rn-Lipschitz-continuous (}Pnpfq ´ Pnpgq}ℓ2pGn´1q ď Rn}f ´ g}ℓ2pGnqq and triviality preserving +(Pnp0q “ 0). For our original node-signal space we also write ℓ2pGq ” ℓ2pG0q. +Graph Convolutional Networks: +A GCN with N layers is then constructed as follows: +Figure 2: Update Rule for a GCN +Let us denote the width of the network at layer n +by Kn. The collection of hidden signals in this +layer can then be thought of a single element of +Ln :“ +à +iPKn +ℓ2pGnq. +(3) +Further let us write the collection of functional +calculus filters utilized to generate the repre- +sentation of this layer by tgn +ijp¨q : 1 ď j ď +Kn´1; 1 ď i ď Knu. Further denoting the char- +acteristic operator of this layer by Tn, the update +rule (c.f. also Fig. 2) from the representation in Ln´1 to Ln is then defined on each constituent in +the direct sum Ln as +f n`1 +i +“ ρn`1 +˜ Kn +ÿ +j“1 +gn`1 +ij +pTn`1qPn`1pf n +j q +¸ +, @1 ď i ď Kn. +We also denote the initial signal space by Lin :“ L0 and the final one by Lout :“ LN. The hence +constructed map from the initial to the final space is denoted by Φ : Lin Ñ Lout. +3 +STABILITY TO INPUT SIGNAL PERTURBATIONS +In order to produce meaningful signal representations, a small input signal change should produce +only a small variation in the output of our GCN. This property is quantified by the Lipschitz constant +of the map Φ associated to the network, which is estimated by our first result below. +Theorem 3.1. With the notation of Section 2 let ΦN : Lin Ñ Lout be the map associated to an +N-layer GCN. We have with Bn :“ +b +supλPσpTnq +ř +jPKn´1 +ř +iPKn |gn +ijpλq|2 for all f, h P Lin that +}ΦNpfq ´ ΦNphq}Lout ď +˜ N +ź +n“1 +LnRnBn +¸ +¨ }f ´ h}Lin +if Tn is normal. For general Tn we have for all tgiju entire, holomorphic and in F hol +ω,C respectively: +Bn :“ +$ +’ +’ +’ +’ +& +’ +’ +’ +’ +% +8ř +k“0 +bř +jPKn´1 +ř +iPKn |pagn +ij qk|2 ¨ }Tn}k +op +bř +jPKn´1 +ř +iPKn }gn +ijp8q}2 ` +1 +2π +ű +BD γT pzq +bř +jPKn´1 +ř +iPKn |gn +ijpzq|2d|z| +bř +jPKn´1 +ř +iPKn }gn +ij}2 +F hol +ω,C +Appendix E contains the corresponding proof and discusses how the derived bound are not necessarily +tight for sparsely connected layers. After Lipschitz constants of connecting operators and non- +linearities are fixed, the stability constant of the network is completely controlled by the tBnu; which +for normal Tn in turn are controlled by the interplay of the utilized filters on the spectrum of Tn. This +allows to combine filters with supλPσpTnq |gn +ijpλq| “ Op1q but supported on complimentary parts of +the spectrum of Tn while still maintaining Bn “ Op1q instead of Op +a +Kn ¨ Kn´1q. In practice one +might thus penalize a ’multiple covering’ of the spectrum by more than one filter at a time during +training in order to increase stability to input signal perturbations. If Tn is not normal but filters are +holomorphic, an interplay persists – with filters now evaluated on a curve and at infinity. +4 + +pn→ 2(Gn) +Pn +2(Gn-1 +(G +l2(Gn) +e2(Gn) +Pn +2(Gn- +2(Gn) +(G)4 +STABILITY TO EDGE PERTURBATIONS +Operators capturing graph-geometries might only be known approximately in real world tasks; e.g. +if edge weights are only known to a certain level of precision. Hence it is important that graph +convolutional networks be insensitive to small changes in the characteristic operators tTnu. Since we +consider graphs with arbitrary vertex weights tµgugPG, we also have to consider the possibility that +these weights are only known to a certain level of precision. In this case, not only do the characteristic +operators Tn, rTn differ, but also the the spaces ℓ2pGq, ℓ2p rGq on which they act. To capture this +setting mathematically, we assume in this section that there is a linear operator J : ℓ2pGq Ñ ℓ2p rGq +facilitating contact between signal spaces (of not-necessarily the same dimension). We then measure +closeness of characteristic operators in the respective spaces by considering the generalized norm- +difference }pJT ´ rTJq}; with J translating between the respective spaces. Before investigating the +stability of entire networks we first comment on single-filter stability. For normal operators we then +find the following result, proved in Appendix A building on ideas first developed in (Wihler, 2009). +Lemma 4.1. Denote by } ¨ }F the Frobenius norm and let T and rT be normal on ℓ2pGq and +ℓ2p rGq respectively. Let g be Lipschitz continuous with Lipschitz constant Dg. For any linear +J : ℓ2pGq Ñ ℓ2p rGq we have }gp rTqJ ´ JgpTq}F ď Dg} rTJ ´ JT}F . +Unfortunately, scalar Lipschitz continuity only directly translates to operator functions if they are +applied to normal operators and when using Frobenius norm (as opposed to e.g. spectral norm). For +general operators we have the following somewhat weaker result, proved in Appendix F: +Lemma 4.2. Let T, rT be operators on on ℓ2pGq , ℓ2p rGq with }T}op, } rT}op ď C. +Let J : +ℓ2pGq Ñ ℓ2p rGq be linear. With Kg “ +1 +2π +ű +BD +1 +|z|γT pzqγ rT pzq|gpzq|d|z| for g holomorphic and +Kg “ ř8 +k“1 |ag +k|kCk´1 for g entire, we have }gpTqJ ´ Jgp rTq}op ď Kg ¨ }JT ´ rTJ}op. +Each Kg itself is interpretable as a semi-norm. For GCNs we find the following (c.f. Appendix F): +Theorem 4.3. Let ΦN, rΦN be the maps associated to N-layer graph convolutional networks with the +same non-linearities and filters, but based on different graph signal spaces ℓ2pGq, ℓ2p rGq, characteristic +operators Tn, rTn and connecting operators Pn, rPn. Assume Bn, rBn ď B as well as Rn, rRn ď R +and Ln ď L for some B, R, L ą 0 and all n ě 0. Assume that there are identification operators +Jn : ℓ2pGnq Ñ ℓ2p rGnq (0 ď n ď N) commuting with non-linearities and connecting operators in +the sense of } rPnJn´1f ´ JnPnf}ℓ2p r +Gnq “ 0 and }ρnpJnfq ´ Jnρnpfq}ℓ2p r +Gnq “ 0. Depending on +whether normal or arbitrary characteristic operators are used, define D2 +n :“ ř +jPKn´1 +ř +iPKn D2 +gn +ij +or D2 +n :“ ř +jPKn´1 +ř +iPKn K2 +gn +ij. Choose D such that Dn ď D for all n. Finally assume that +}JnTn ´ rTnJn}˚ ď δ and with ˚ “ F if both operators are normal and ˚ “ op otherwise. Then we +have for all f P Lin and with Jn the operator that the Kn copies of Jn induce through concatenation +that }rΦpJ0fq ´ JNΦpfq} Ă +Lout ď N ¨ DRL ¨ pBRLqN´1 ¨ }f}Lin ¨ δ. +The result persists with slightly altered constants, if identification operators only almost commute with +non-linearities and/or connecting operators, as Appendix G further elucidates. Since we estimated +various constants (Bn, Dn, ...) of the individual layers by global ones, the derived stability constant +is clearly not tight. However it portrays requirements for stability to edge level perturbations well: +While the (spectral) interplay of Section 3 remains important, it is now especially large single-filter +stability constants in the sense of Lemmata 4.1 and 4.2 that should be penalized during training. +5 +STABILITY TO STRUCTURAL PERTURBATIONS: TRANSFERABILITY +While the demand that } rTJ ´ JT} be small in some norm is well adapted to capture some notions +of closeness of graphs and characteristic operators, it is too stringent to capture others. As an +illustrative example, further developed in Section 5.2 and numerically investigated in Section 7 below, +suppose we are given a connected undirected graph with all edge weights of order Op1{δq. With the +Laplacian as characteristic operator (governing heat-flow in Physics (Cole, 2011)), we may think +of this graph as modelling an array of coupled heat reservoirs with edge weights corresponding to +5 + +heat-conductivities. As 1{δ Ñ 8, the conductivities between respective nodes tend to infinity, heat +exchange is instantaneous and all nodes act as if they are fused together into a single large entity – with +the graph together with its characteristic operator behaving as an effective one-dimensional system. +This ’convergent’ behaviour is however not reflected in our characteristic operator, the graph Laplacian +∆δ: Clearly }∆δ}op “ 1{δ ¨ }∆1}op Ñ 8 as 1{δ Ñ 8. Moreover, we would also expect a Cauchy- +like behaviour from a ’convergent system’, in the sense that if we for example keep 1{δa ´ 1{δb “ 1 +constant but let p1{δaq, p1{δbq Ñ 8 we would expect }∆δa ´ ∆δb}op Ñ 0 by a triangle-inequality +argument. However, we clearly have }∆δa ´∆δb}op “ |1{δa ´1{δb|¨}∆1}op “ }∆1}op, which does +not decay. The situation is different however, when considering resolvents of the graph Laplacian. +An easy calculation (c.f. Appendix H) yields }pωId ´ ∆δbq´1 ´ pωId ´ ∆δaq´1}op “ Opδa ¨ δbq +so that we recover the expected Cauchy behaviour. What is more, we also find the convergence +pωId ´ ∆δq´1 Ñ P0 ¨ pω ´ 0q´1; where P0 denotes the projection onto the one-dimensional lowest +lying eigenspace of the ∆δs (spanned by the vectors with constant entries). We may interpret pω´0q´1 +as the resolvent of the graph Laplacian of a singleton (since such a Laplacian is identically zero) and +thus now indeed find our physical intuition about convergence to a one-dimensional system reflected +in our formulae. Motivated by this example, Section 5.1 develops a general theory for the difference +in outputs of networks evaluated on graphs for which the resolvents Rω :“ pωId ´ Tq´1 and +rRω :“ pωId ´ rTq´1 of the respective characteristic operators are close in some sense. Subsequently, +Section 5.2 then further develops our initial example while also considering an additional setting. +5.1 +GENERAL THEORY +Throughout this section we fix a complex number ω P C and for each operator T assume ω, ω R σpTq. +This is always true for ω with |ω| ě }T}op, but if T is additionally self adjoint one could set ω “ i. +If T is non-negative one might choose ω “ p´1q). As a first step, we then note that the conclusion of +Lemma 4.1 can always be satisfied if we chose J ” 0. To exclude this case – where the application +of J corresponds to losing too much information – we follow Post (2012) in making the following +definition: +Definition 5.1. Let J : ℓ2pGq Ñ ℓ2p rGq and rJ : ℓ2p rGq Ñ ℓ2pGq be linear, and let T ( rT) be operators +on (ℓ2pGq) (ℓ2p rGq). We say that J and rJ are ϵ-quasi-unitary with respect to T, rT and ω if +}Jf}ℓ2p r +Gq ď 2}f}ℓ2pGq, +}pJ ´ rJ˚qf}ℓ2p r +Gq ď ϵ}f}ℓ2pGq, +}pId ´ rJJqRωf}ℓ2pGq ď ϵ}f}ℓ2pGq, +}pId ´ J rJq rRωu}ℓ2p r +Gq ď ϵ}u}ℓ2p r +Gq. +(4) +The motivation to include the resolvents in the norm estimates (4) comes from the setting where +T “ ∆ is the graph Laplacian and ω “ p´1q. In that case, the left equation in (4 is for example +automatically fulfilled when demanding }pId ´ rJJqf}2 +ℓ2pGq ď ϵp}f}2 ` E∆pfqq +1 +2 , with E∆p¨q “ +x¨, ∆¨yℓ2pGq the (positive) energy form induced by the Laplacian ∆ (Post, 2012). This can thus be +interpreted as a relaxation of the standard demand }pId ´ rJJq}op ď ϵ. Relaxing the demands of +Section 4, we now demand closeness of resolvents instead of closeness of operators: +Definition 5.2. If, for ω P C and linear J : ℓ2pGq Ñ ℓ2p rGq the resolvents Rω and rRω satisfy +}p rRωJ ´ JRωqf}ℓ2p r +Gq ď ϵ}f}ℓ2pGq for all f P ℓ2pGq, T and rT are called ω-ϵ-close with identifica- +tion operator J. If additonally }p rR˚ +ωJ ´ JR˚ +ωqf}ℓ2p r +Gq ď ϵ}f}ℓ2pGq, they are doubly ω-ϵ-close. +Our first result establishes that operators being (doubly-)ω-ϵ-close indeed has useful consequences: +Lemma 5.3. Let T ( rT) be operators on ℓ2pGq (ℓ2p rGq). If these operators are ω-ϵ-close with +identification operator J, and }Rω}op, } rRω}op ď C we have }JgpTq ´ gp rTqJ}op ď Kg ¨ }p rRωJ ´ +JRωq}op with Kg “ +1 +2π +ű +BDp1 ` |z ´ ω|γT pzqqp1 ` |z ´ ω|γ rT pzqq|gpzq|d|z| for holomorphic g, +Kg “ }g}F hol +ω,C if g P F hol +ω,C and Kg “ }g}F cont +ω,C for T, rT normal and doubly ω-ϵ-close. +This result may then be extended to entire networks, as detailed in Theorem 5.4 below whose +statement persists with slightly altered stability constants, if identification operators only almost +commute with non-linearities and/or connecting operators. Proofs are contained in Appendix I. +Theorem 5.4. Let ΦN, rΦN be the maps associated to N-layer graph convolutional networks with +the same non-linearities and functional calculus filters, but based on different graph signal spaces +6 + +ℓ2pGnq, ℓ2p rGnq, characteristic operators Tn, rTn and connecting operators Pn, rPn. Assume Bn, rBn ď +B as well as Rn, rRn ď R and Ln ď L for some B, R, L ą 0 and all n ě 0. Assume that +there are identification operators Jn : ℓ2pGnq Ñ ℓ2p rGnq (0 ď n ď N) commuting with non- +linearities and connecting operators in the sense of } rPnJn´1f ´JnPnf}ℓ2p r +Gnq “ 0 and }ρnpJnfq´ +Jnρnpfq}ℓ2p r +Gnq “ 0. define D2 +n :“ ř +jPKn´1 +ř +iPKn K2 +gn +ij with Kgn +ij as in Lemma 5.3. Choose D +such that Dn ď D for all n. Finally assume that }JnpωId ´ Tnq´1 ´ pωId ´ rTnq´1Jn}op ď ϵ. If +filters in F cont +ω,C are used, assume additionally that }JnppωId´Tnq´1q˚´ppωId´ rTnq´1q˚Jn}op ď ϵ. +Then we have for all f P Lin and with Jn the operator that the Kn copies of Jn induce through +concatenation that }rΦNpJ0fq ´ JNΦNpfq} Ă +Lout ď N ¨ DRL ¨ pBRLqN´1 ¨ }f}Lin ¨ ϵ. +5.2 +EXEMPLARY APPLICATIONS +Collapsing Strong Edges: +We first pick our example from the beginning of section 5 up again +and generalize it significantly: We now consider the graph that we collapse to a single node to be a +sub-graph (of strong edges) embedded into a larger graph. Apart from coupled heat reservoirs, this +setting also e.g. captures the grouping of close knit communities within social networks into single +entities, the scale-transition of changing the description of (the graph of) a molecule from individual +atoms interacting via the coulomb potential Z1Z2{R (with R the distance and Z1, Z2 atomic charges) +to the interaction of (functional) groups comprised of closely co-located atoms, or spatial networks if +weights are set to e.g. inverse distances. In what follows, we shall consider two graphs with vertex +sets G and rG. We consider G to be a subset of the vertex set rG and think of the graph corresponding +to G as arising in a collapsing procedure from the ’larger’ graph rG. +More precisely, we assume that the vertex set rG can be split into +three disjoint subsets rG “ rGLatin +Ť rGGreek +Ťt‹u (c.f. also Fig. 3). +We assume that the adjacency matrix Ă +W when restricted to Latin +vertices or a Latin vertex and the exceptional node ’‹’ is of order +unity p Ą +Wab, Ă +Wa‹ “ Op1q, @a, b P rGLatinq. For Greek indices, we +assume that we may write Ă +Wαβ “ ωαβ +δ +and Ă +Wα‹ “ ωα‹ +δ +such that +pωαβ, ωα‹ “ Op1q for all α, β P rGGreek. We also assume that the +sub-graph corresponding to vertices in rGGreek +Ťt‹u is connected. +We then take G “ rGLatin +Ťt‹u (c.f. again Fig. 3). The adjacency ma- +trix W on this graph is constructed by defining Wab “ Ă +Wab, @a, b P +rGLatin and setting (with Wa‹ ” W‹a) +W‹a :“ Ă +Wa‹ ` +ÿ +βP r +GGreek +Ă +Waβ +´ +@a P rGLatin +¯ +. +We also allow our graph rG to posses node-weights trµrgurgP r +G that are +not necessarily equal to one. The Laplace operator ∆ r +G acting on the +graph signal space ℓ2p rGq induces a positive semi-definite and convex +Figure 3: Collapsed (left) and +original (right) Graphs +energy form on this signal space via E r +Gpuq :“ xu, ∆ r +Guyℓ2p r +Gq “ ř +g,hP r +G Ă +Wgh|upgq´uphq|2. Using +this energy form, we now define a set comprised of |G| signals, all of which live in ℓ2p rGq. These +signals are used to facilitate contact between the respective graph signal spaces ℓ2pGq and ℓ2p rGq. +Definition 5.5. For each g P G, define the signal ψδ +g P ℓ2p rGq as the unique solution to the convex +optimization program +min E r +Gpuq subject to uphq “ δhg for all h P rGLatin +ď +t‹u. +(5) +Given the boundary conditions, what is left to determine in the above optimization program are the +’Greek entries’ ψδ +gpαq of each ψδ +g. As Appendix J further elucidates, these can be calculated explicitly +and purely in terms of the inverse of ∆ r +G restricted to Greek indices as well as (sub-)columns of the +adjacency matrix Ă +W. Node-weights on G are then defined as µδ +g :“ ř +hP r +G ψδ +gphq ¨ rµh. We denote +7 + +the corresponding signal space by ℓ2pGq. Importantly, one has µδ +a Ñ rµa for any Latin index and +µδ +‹ Ñ rµ‹ ` ř +αP r +GGreek rµα as δ Ñ 0; which recovers our physical intuition about heat reservoirs. To +translate signals from ℓ2pGq to ℓ2p rGq and back, we define two identification operators J : ℓ2pGq Ñ +ℓ2p rGq and rJ : ℓ2p rGq Ñ ℓ2pGq via Jf :“ ř +gPG fpgq ¨ ψδ +g and p rJuqpgq :“ xu, ψδ +gyℓ2p r +Gq{µδ +g for all +f P ℓ2pGq, u P ℓ2p rGq and g P G. Our main theorem then states the following: +Theorem 5.6. With definitions and notation as above, there are constants K1, K2 ě 0 such that the +operators J and rJ are pK1 +? +δq-quasi-unitary with respect to ∆ r +G, ∆G and ω “ p´1q. Furthermore, +the operators ∆ r +G and ∆G are p´1q-pK2 +? +δq close. with identification operator J. +Appendix J presents the (fairly involved) proof of this result. Importantly, the size of the constants +K1, K2 is independent of the cardinality (or more precisely the total weight) of rGLatin, implying that +Theorem 5.6 also remains applicable in the realm of large graphs. Finally we note, that this stability +result is contingent on the use of the (un-normalized) graph Laplacian (c.f. Appendix K): +Theorem 5.7. In the setting of Theorem 5.6 denote by T ( rT) adjacency matrices or normalized +graph Laplacians on ℓ2pGq (ℓ2pGq). There are no functions η1, η2 : r0, 1s Ñ Rě0 with ηipδq Ñ 0 +as δ Ñ 0 (i “ 1, 2), families of identification operators Jδ, rJδ and ω P C so that Jδ and rJδ are +η1pδq-quasi-unitary with respect to rT, T and ω while the operators rT and T remain ω-η2pδq close. +The Realm of Large Graphs: +In order to relate our transferability +framework to the literature, we consider an ’increasing’ sequence +of graphs (Gn Ď Gn`1) approximating a limit object, so that the +transferability framework of Levie et al. (2019a) is also applicable. +We choose the limit object to be the circle of circumference 2π and +our approximating graphs to be the closed path-graph on N vertices Figure 4: Closed Path-Graphs +equidistantly embedded into the circle (c.f. Fig 4). With h “ 2π{N the node-distance, we set weights +to 1{h2; ensuring consistency with the ’continuous’ Laplacian in the limit N Ñ 8. More details are +presented in Appendix L, which also contains the proof of the corresponding transferability result: +Theorem 5.8. In the above setting choose all node-weights equal to one and N to be odd for +definiteness. There exists constants K1, K2 “ Op1q so that for each N ě 1, there exist identification +operators J, rJ mapping between ℓ2pGNq and ℓ2pGN`1q so that J and rJ are pK1{Nq-quasi-unitary +with respect to ∆GN , ∆GN`1 and ω “ p´1q. Furthermore, the operators ∆GN and ∆GN`1 are +p´1q-pK2{Nq close with identification operator J. +Lemma 5.3 then implies an Op 1 +N q-decay of }gpTqJ ´ Jgp rTq}op for fixed g. This reduces to an +Op +? +N +N q-decay for Levie et al. (2019a) (ibid. Theorem 5, pt. 3) assuming a similar decay of operator- +distances. Our framework might this capture transferability properties other approaches could miss. +6 +GRAPH LEVEL STABILITY +To solve tasks such as graph classification or regression over multiple graphs, graphs of varying sizes +need to be represented in a common feature space. Here we show that aggregating node-level features +into such graph level features via p-norms (}f}ℓppGq :“ př +gPG |fg|pµgq1{p) preserves stability. To +Figure 5: Graph Level Aggregation +this end, let Lout be a target space of a GCN in the sense of (3). +On each of the (in total Kout) ℓ2pGoutq summands of Lout, we +may apply the map fi ÞÑ }fi}ℓppGoutq. Stacking these maps, we +build a map from Lout to RKout. Concatenating the map ΦN +associated to an N-layer GCN with this map yields a map from +Lin to RKout. We denote it by Ψp +N and find: +Theorem 6.1. For p ě 2 we have in the setting of Theorem 3.1 that }Ψp +Npfq ´ Ψp +Nphq}RKout ď +´śN +n“1 LnRnBn +¯ +¨ }f ´ h}Lin. In the setting of Theorem 4.3 or 5.4 and under the additional +assumption that the ’final’ identification operator JN satisfies +ˇˇ}JNfi}ℓkp r +GNq ´ }fi}ℓkpGNq +ˇˇ ď +δ ¨ K ¨ }fi}ℓ2pGNq for all fi P ℓ2pGNq, we have }Ψp +Npfq ´ rΨp +NpJ0fq}RKout ď pN ¨ DRL ` K ¨ +pBRLqq ¨ pBRLqN´1 ¨ }f}Lin ¨ δ. +8 + +/· Ilep(Gout) R +l2(Gin)— +.ep(Gout) +C→l2 (Gout) +R +l2(Gin)-N +.ep(Gout) +→l2(Gout) +RDerived stability results thus persist (under mild assumptions) if graph level features are aggregated +via p-norms. Appendix M contains the corresponding proof. +7 +NUMERICAL RESULTS +We focus on investigating structural perturbations, as correspond- +ing results are most involved and novel: +We first consider a graph on 5 nodes with an adjacency matrix A +with Op1q-entries (c.f. 30 in Appendix N). We then scale A by +1{δa and 1{δb (with +1 +δa ´ 1 +δb “ 1) respectively and consider the +norm-difference between associated Laplacians and resolvents. +Fig. 6 (a) then illustrate the theoretical result (c.f. Section 5) that +resolvent- instead of Laplacian-differences capture the conver- +gence behaviour. Embedding the considered graph into a larger +graph (Ă +W P R8ˆ8; c.f. (31) in Appendix N), we consider the +collapsing edge setting of Section 5.2 in Fig. 6 (b). As expected, +the corresponding resolvents do approach each other as δ Ñ 0. +Contrary to the theoretical bound in Lemma 5.3, differences of +resolvent-monomials decrease as their power k increases. +Beyond small graphs – inaccessible to traditional asymptotic +methods – our method is also applicable to the large-graph setting: +Fig. 7 picks up the example of an ’increasing’ graph sequence +’approximating’ the circle again. As predicted in Section 5.2, the +difference in resolvents decays (9 1 +N ). Fig. 10 in Appendix N +shows how the difference in Laplacians diverges instead. Hence Figure 6: Edge-Collapse Stability +Figure 7: The Large-N Regime +our framework might capture stability properties traditional ap- +proaches could miss. +Finally, we investigate the transferability of a two-layer GCN +with 16 nodes per hidden Layer combined with the aggregation +method of Section 6 into a graph-level map Ψp +2. Filters are of +the form (2) up to order k “ 11. Coefficients tbg +ku are sampled +uniformly from r´100, 100s. Feature vectors are generated on +the QM7 dataset. There each graph represents a molecule; nodes +correspond to individual atoms. Adjacency matrices are given by +Ă +Wij “ ZiZj{}xi ´ xj} with Zi (xi) the atomic charge (equilib- +rium position) of atom i. We choose node-weights as rµi “ Zi +and the Laplacian as characteristic operator. Leading up to Fig. 8 +we consider the graph of methane (5 Nodes; one Carbon (Z1 “ 6) +and four Hydrogen nodes (Zią1 “ 1)) and deflect one of the Hy- +drogen atoms (i “ 2) out of equilibrium and along a straight line +towards the Carbon atom. We then consider the transferability of +the entire GCN between the resulting graph and an effective graph +combining Carbon and deflected Hydrogen into a single node +"‹" with weight µ‹ “ Z1 ` Z2 “ 7 located at the equilibrium +position of Carbon. With J translating from effective to original +description, we consider }Ψp +2pfq ´ Ψp +2pJfq}R16 (averaged over +100 random unit-norm choices of f) as a function of }x1 ´x2}´1. +At equilibrium the transferability error is Op1q. It decreases fast +with decreasing Carbon-Hydrogen distance, with the choice of +Figure 8: GCN Transferability +representation (effective vs. original) quickly becoming insignificant for generated feature vectors. +8 +DISCUSSION +A theoretically well founded framework capturing stability properties of GCNs was developed. We +related node-level stability to (spectral) covering properties and edge-level stability to introduced +semi-norms of employed filters. For non-normal characteristic operators, tools from complex analysis +provided grounds for derived stability properties. We introduced a new notion of stability to structural +9 + +102 +II sa -△s ll op +101 +100 +10-1 +ResolventDifferences +Operator Differences +10-2 +10-3 +R-1(s）)- R-1(s)lop +10-4 +10-5 +0 +20 +40 +1/8a +60 +80 +100 +a +k=1 +10-2 +IlRk,J - JRillop +k=2 +k=3 +10-3 +k=4 +k=5 +10-4 +10-5 +10-6 +10-7 +10-8 +10-9 +(b) +0 +20 +40 +1/8 +60 +80 +10010-2 +10-3 +Resolvent Differences +R-1(△GN+1)J- JR-1(△G)lop +10-4 +10-5 +10-6 +10-7 +10-8 +0 +250 +500 +750 +1000 +1250 +1500 +1750 +2000 +N蚂(Jf)－() +p= +2 +R16 +p += +3 +p= 4 +100 +p = 5 +p = 6 +p += +7 +p-Norm Differences +10-1 +10-2 +10-3 +0 +20 +40 +60 +80 +100 +equilibrium distancel +/-1 +C1perturbations, highlighted the importance of the resolvent and detailed how the developed line of +thought captures relevant settings of structural changes such as the collapse of a strongly connected +sub-graph to a node. 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URL +https://arxiv.org/abs/2111.14522. +Thomas Wiatowski and Helmut Bölcskei. A mathematical theory of deep convolutional neural +networks for feature extraction. IEEE Transactions on Information Theory, 64:1845–1866, 2018. +T.P. Wihler. On the hölder continuity of matrix functions for normal matrices. Journal of inequalities +in pure and applied mathematics, 10(4), Dec 2009. ISSN 1443-5756. URL https://www. +emis.de/journals/JIPAM/images/276_09_JIPAM/276_09_www.pdf. +11 + +A +SOME CONCEPTS IN LINEAR ALGEBRA +In the interest of self-containedness, we provide a brief review of some concepts from linear algebra +utilized in this work that might potentially be considered more advanced. Presented results are all +standard; a very thorough reference is Michael Reed (1981). +Hilbert Spaces: +To us, a Hilbert space — often denoted by H — is a vector space over the complex +numbers which also has an inner product — often denoted by x¨, ¨yH. Prototypical examples are +given by the Euclidean spaces Cd with inner product xx, yyCd :“ řd +i“1 xiyi. Associated to an inner +product is a norm, denoted by } ¨ }H and defined by }x}H :“ +a +xx, xyH for x P H. +Direct Sums of Spaces: +Given two potentially different Hilbert spaces H and p +H, one can form +their direct sum H ‘ p +H. Elements of H ‘ p +H are vectors of the form pa, bq, with a P H and b P p +H. +Addition and scalar multiplication are defined in the obvious way by +pa, bq ` λpc, dq :“ pa ` λc, b ` λdq +for a, c P H, b, d P p +H and λ P C. The inner product on the direct sum is defined by +xpa, bq, pc, dqyH‘ p +H :“ xa, cyH ` xb, dy p +H. +As is readily checked, this implies that the norm } ¨ }H‘ p +H on the direct sum is given by +}pa, bq}2 +H‘ p +H :“ }a}2 +H ` }b}2 +p +H. +Standard examples of direct sums are again the Euclidean spaces, where one has Cd “ Cn ‘ Cm if +m ` n “ d, as is easily checked. One might also consider direct sums with more than two summands, +writing Cd “ ‘d +i“1C for example. In fact, one might also consider infinite sums of Hilbert spaces: +The space ‘8 +i“1Hi is made up of those elements a “ pa1, a2, a3, ...q with ai P Hi for which the +norm +}a}2 +‘8 +i“1Hi :“ +8 +ÿ +i“1 +}ai}2 +Hi +is finite. This means for example that the vector p1, 0, 0, 0, ...q is in ‘8 +i“1C, while p1, 1, 1, 1, ...q is +not. +Direct Sums of Maps: +Suppose we have two collections of Hilbert spaces tHiuΓ +i“1, t r +HiuΓ +i“1 with +Γ P N or Γ “ 8. Suppose further that for each i ď Γ (resp. i ă Γ) we have a (not necessarily linear) +map Ji : Hi Ñ r +Hi. Then the collection tJiuΓ +i“1 of these ’component’ maps induce a ’composite’ +map +J : ‘Γ +i“1Hi ÝÑ ‘Γ +i“1 r +Hi +between the direct sums. Its value on an element a “ pa1, a2, a3, ...q P ‘Γ +i“1Hi is defined by +J paq “ pJ1pa1q, J2pa2q, J3pa3q, ...q P ‘Γ +i“1 r +Hi. +Strictly speaking, one has to be a bit more careful in the case where Γ “ 8 to ensure that +}J paq}‘8 +i“1 r +Hi ‰ 8. This can however be ensured if we have }Jipaiq} r +Hi ď C}ai}Hi for all +1 ď i and some C independent of all i, since then }J paq}‘8 +i“1 r +Hi ď C}a}‘8 +i“1Hi ď 8. If each Ji is +a linear operator, such a C exists precisely if the operator norms (defined below) of all Ji are smaller +than some constant. +Operator Norm: +Let J : H Ñ r +H be a linear operator between Hilbert spaces. We measure its +’size’ by what is called the operator norm, denoted by } ¨ }op and defined by +}J}op :“ +sup +ψPH,}ψ}H“1 +}Aψ} r +H +}ψ}H +. +12 + +Adjoint Operators +Let J : H Ñ r +H be a linear operator from the Hilbert space H to the Hilbert +space r +H. Its adjoint J˚ : r +H Ñ H is an operator mapping in the opposite direction. It is uniquely +determined by demanding that +xJf, uy r +H “ xf, J˚uyH +holds true for arbitrary f P H and u P r +H. +Normal Operators: +If a linear operator ∆ : H Ñ H maps from and to the same Hilbert space, +we can compare it directly with its adjoint. If ∆∆˚ “ ∆˚∆, we say that the operator ∆ is normal. +Special instances of normal operators are self-adjoint operators, for which we have the stronger +property ∆ “ ∆˚. If an operator is normal, there are unitary maps U : H Ñ H diagonalizing ∆ as +U ˚∆U “ diagpλ1, ...λnq, +with eigenvalues in C. We call the collection of eigenvalues the spectrum σp∆q of ∆. If dim H “ d, +we may write σp∆q “ tλud +i“1. It is a standard exercise to verify that each eigenvalue satisfies +|λi| ď }∆}op. Associated to each eigenvalue is an eigenvector φi. The collection of all (normalized) +eigenvectors forms an orthonormal basis of H. We may then write +∆f “ +dÿ +i“1 +λi xφi, fyHφi. +Resolvent of an Operator: +Given an operator T on some Hilbert space H, we have by definition +that the operator pT ´ zq : H Ñ H is invertible precisely if z ‰ σpTq. In this case we write +RzpTq “ pzId ´ Tq´1 +and call this operator the resolvent of T at z. +If T is normal it can be proved that the norm of the resolvent satisfies +}RzpTq}op “ +1 +distpz, σp∆qq, +where distpz, σp∆qq denotes the minimal distance between z and any eigenvalue of ∆. For non- +normal operators, one can prove +}RzpTq}op ď γT pzq +with +γT pzq “ exp r2}T}1{dpz, σpTqqs {dpz, σpTqq +as is proved in Bandtlow (2004a). +Frobenius Norm: +Given two finite dimensional Hilbert spaces H1 and H2 with orthonormal bases +tφ1 +i ud1 +i“1 and tφ1 +i ud1 +i“1, the Frobenius norm } ¨ }F of an operator A : H1 Ñ H2 may be defined as +}A}2 +2 :“ +d2 +ÿ +i“1 +d1 +ÿ +j“1 +|Aij|2 +with Aij the matrix representation of A with respect to the bases tφ1 +i ud1 +i“1 and tφ1 +i ud1 +i“1. It is a +standard exercise to verify that this norm is indeed independent of any choice of basis and hence +invariant under multiplying A with a unitary on either the left or the right side. More precisely, if +U : H2 Ñ H2 and V : H1 Ñ H1 are unitary, we have +}UAV }2 +F “ }A}2 +F . +Frobenius norms can be used to transfer Lipschitz continuity properties of complex functions to the +setting of functions applied to normal operators: +Lemma A.1. Let g : C Ñ C be Lipschitz continuous with Lipschitz constant Dg. This implies +}gpXqJ ´ JgpY q}F ď Dg ¨ }X ´ Y }F . +for normal operators X on H2, Y on H1 and any linear map J : H1 Ñ H2. +13 + +Proof. This proof is a modified version of the proof in Wihler (2009). Let U, W be unitary (with +respect to the inner product x¨, ¨yH) operators diagonalizing the normal operators X and Y as +V ˚XV “ diagpλ1, ...λd2q “: DpXq +W ˚Y W “ diagpµ1, ...µd1q “: DpY q. +Since the Frobenius norm is invariant under unitary transformations we find +}gpXqJ ´ JgpY q||2 +F “ ||gpV DpXqV ˚q ´ gpWDpY qW ˚q}2 +F +“ }V gpDpXqqV ˚J ´ JWgpDpY qqW ˚}2 +F +“ }gpDpXqqV ˚JW ´ V ˚JWgpDpY qq}2 +F +“ +ÿ +i,j +|pgpDpXqqV ˚JW ´ V ˚JWgpDpY qqqij|2 +“ +ÿ +i,j +ˇˇˇˇˇ +ÿ +k +rgpDpXqqsikrV ˚JWskj ´ rV ˚JWsikrgpDpY qqskj +ˇˇˇˇˇ +2 +“ +ÿ +i,j +|rV ˚Wsij|2 |gpλjq ´ gpµiq|2 +ď +ÿ +i,j +|rV ˚Wsij|2 D2 +g|λj ´ µi|2 +“ D2 +g}X ´ Y }2 +F . +B +APPROXIMATING BOUNDED CONTINUOUS FILTERS +Let us recall Definition 2.1: +Definition B.1. Fix ω P C and C ą 0. Define the space F cont +ω,C of continuous filters on Cztω, ωu, +to be the space of multilinear power-series’ gpzq “ ř8 +µ,ν“0 aµν pω ´ zq´µ pω ´ zq´µ for which the +norm }g}F cont +ω,C :“ ř8 +µ,ν“0 |µ ` ν|Cµ`ν|aµν| is finite. +We now prove that upon denoting by Bϵpωq Ď C the open ball of radius ϵ around ω, one can show +that for arbitrary δ ą 0 and every continuous function g defined on CzpBϵpωq Y Bϵpωqq which is +regular at infinity – i.e. satisfies limrÑ`8 gprzq “ c P C independent of which z ‰ 0 is chosen – +there is a function f P F cont +ω,C so that |fpzq ´ gpzq| ď δ for all z P CzpBϵpωq Y Bϵpωqq. +Making use of the Stone-Weierstrass theorem for complex functions, it suffices to prove that for every +point z in CzpBϵpωq Y Bϵpωqq there are functions f and g in F cont +ω,C for which +fpzq ‰ gpzq. +But this is obvious since pω ´ zq´1 is injective on CzpBϵpωq Y Bϵpωqq. +C +COMPLEX ANALYSIS +A general reference for topics discussed in this section is Bak & Newman (2017). +For a complex valued function f of a single complex variable, the derivative of f at a point z0 P C in +its domain of definition is defined as the limit +f 1pz0q :“ lim +zÑz0 +fpzq ´ fpz0q +z ´ z0 +. +For this limit to exist, it needs to be independent of the ’direction’ in which z approaches z0, which is +a stronger requirement than being real-differentiable. A function is called holomorphic on an open set +U if it is complex differentiable at every point in U. It is called entire if it is complex differentiable at +every point in C. Every entire function has an everywhere convergent power series representation +gpzq “ +8 +ÿ +k“0 +agzk. +(6) +14 + +If a function g is analytic (i.e. can be expanded into a power series), we have +gpλq “ ´ 1 +2πi +¿ +S +gpzq +λ ´ z dz +(7) +for any circle S Ď C encircling λ by Cauchy’s integral formula. +In fact, the integration contour need not be a circle S, but may be the boundary of any so called +Cauchy domain containing λ: +Definition C.1. A subset D of the complex plane C is called a Cauchy domain if D is open, has a +finite number of components (the closure of two of which are disjoint) and the boundary of BD of D +is composed of a finite number of closed rectifiable Jordan curves, no two of which intersect. +Equation (7) forms the backbone of complex analysis. Since the integral +I :“ ´ 1 +2πi +¿ +BD +gpzqpzId ´ Tq´1dz +(8) +is well defined for holomorphic gp¨q and any operator T for which σpTq and BD are disjoint (c.f. e.g. +Post (2012) for details), we can essentially take (8) as a defining equation through which one might +apply holomorphic functions to operators. +While functions that are everywhere complex differentiable have a series representation according +to (6), complex functions that are holomorphic only on Cztωu have a series representation (called +Laurent series) according to +gpzq “ +8 +ÿ +k“´8 +akpz ´ ωqk. +If these functions are assumed to be regular at infinity, no terms with positive exponent are permitted +and (changing the indexing) we may thus write +gpzq “ +8 +ÿ +k“0 +akpz ´ ωq´k. +Motivated by this, we now prove the following consistency result: +Lemma C.2. With the notation of Section 2 we have for any k ě 1 and ω R σpTq that +pω ¨ Id ´ Tq´k :“ +1 +2πi +¿ +BD +pω ´ zq´k ¨ pzId ´ Tq´1dz, +where we interpret the left hand side of the equation in terms of inversion and matrix powers. +Proof. We first note that we may write +RλpTq “ +8 +ÿ +n“0 +pλ ´ ωqnp´1qnRωptqn`1 +for |λ ´ ω| ď }RωpTq} using standard results in matrix analysis (namely the ’Neumann Characteri- +sation of the Resolvent’ which is obtained by repeated application of a resolvent identity; c.f. Post +(2012) for more details). We thus find +1 +2πi +¿ +BD +ˆ +1 +ω ´ z +˙k +1 +zId ´ T dz “ +1 +2πi +¿ +BD +ˆ +1 +ω ´ z +˙k +8 +ÿ +n“0 +pω ´ zqnRωpTqn`1. +Using the fact that +1 +2πi +¿ +BD +pz ´ ωqn´k´1dz “ δnk +then yields the claim. +15 + +D +PROOF OF LEMMA 2.3 +We want to prove the following: +Lemma +D.1. +For holomorphic +g +and generic +T +we have +}gpTq}op +ď +|gp8q| ` +1 +2π +ű +BD |gpzq|γT pzqd|z|. Furthermore we have for any T with γT pωq ď C, that }gpTq}op ď }g}F hol +ω,C +as long as g P FC,ω. +Proof. We first note +›››››› +gp8q ¨ Id ` +1 +2πi +¿ +BD +gpzq ¨ pzId ´ Tq´1dz +›››››› +op +ď }gp8q ¨ Id}op ` +›››››› +1 +2πi +¿ +BD +gpzq ¨ pzId ´ Tq´1dz +›››››› +op +ď |gp8q| ` 1 +2π +¿ +BD +|gpzq| +››¨pzId ´ Tq´1›› +op d|z|. +The first claim thus follows together with }RzpTq}op ď γT pzq. The second claim can be derived as +follows: +}gpTq}op “ +››››› +8 +ÿ +k“0 +bg +kpT ´ ωq´k +››››› +op +ď +8 +ÿ +k“0 +|bg +k| +››pT ´ ωq´k›› +op ď +8 +ÿ +k“0 +|bg +k|γT pωqk ď +8 +ÿ +k“0 +|bg +k|Ck. +E +PROOF OF THEOREM 3.1 AND TIGHTNESS OF RESULTS +. We want to prove the following: +Theorem E.1. With the notation of Section 2 let ΦN : Lin Ñ Lout be the map associated to an +N-layer GCN. We have +}ΦNpfq ´ ΦNphq}Lout ď +˜ N +ź +n“1 +LnRnBn +¸ +¨ }f ´ h}Lin +with Bn :“ +b +supλPσpTnq +ř +jPKn´1 +ř +iPKn |gn +ijpλq|2 if Tn is normal. For general Tn we have for all +tgiju entire, holomorphic and in Fω,C respectively: +Bn :“ +$ +’ +’ +’ +’ +& +’ +’ +’ +’ +% +8ř +k“0 +bř +jPKn´1 +ř +iPKn |pagn +ij qk|2 ¨ }Tn}k +op +bř +jPKn´1 +ř +iPKn }gn +ijp8q}2 ` +1 +2π +ű +Γ γT pzq +bř +jPKn´1 +ř +iPKn |gn +ijpzq|2d|z| +bř +jPKn´1 +ř +iPKn }gn +ij}2 +ω,C +Proof. Given input signals f, hn P Lin, let us – sticking to the notation introduced in Section 2 – +denote the intermediate signal representations in the intermediate layers Ln by f n, hn P Ln. With +the update rule described in Section 2 and the norm induced on each Ln as described in Appendix A, +we then have +}f n`1 ´ hn`1}2 +Ln`1 +“ +Kn`1 +ÿ +i“1 +›››››ρn`1 +˜ Kn +ÿ +j“1 +gn`1 +ij +pTn`1qPn`1pf n +j q +¸ +´ ρn`1 +˜ Kn +ÿ +j“1 +gn`1 +ij +pTn`1qPn`1phn +j q +¸››››› +2 +ℓ2pGn`1q +ďL2 +n`1 +Kn`1 +ÿ +i“1 +››››› +Kn +ÿ +j“1 +gn`1 +ij +pTn`1qPn`1pf n +j q ´ +Kn +ÿ +j“1 +gn`1 +ij +pTn`1qPn`1phn +j q +››››› +2 +ℓ2pGn`1q +“L2 +n`1 +Kn`1 +ÿ +i“1 +››››› +Kn +ÿ +j“1 +gn`1 +ij +pTn`1q +“ +Pn`1pf n +j q ´ Pn`1phn +j q +‰ +››››› +2 +ℓ2pGn`1q +. +16 + +We next note +Kn`1 +ÿ +i“1 +››››› +Kn +ÿ +j“1 +gn`1 +ij +pTn`1q +“ +Pn`1pf n +j q ´ Pn`1phn +j q +‰ +››››› +2 +ℓ2pGn`1q +ď +Kn`1 +ÿ +i“1 +˜ Kn +ÿ +j“1 +}gn`1 +ij +pTn`1q}op} +“ +Pn`1pf n +j q ´ Pn`1phn +j q +‰ +}ℓ2pGn`1q +¸2 +ď +˜Kn`1 +ÿ +i“1 +Kn +ÿ +j“1 +}gn`1 +ij +pTn`1q}2 +op +¸ Kn +ÿ +j“1 +}} +“ +Pn`1pf n +j q ´ Pn`1phn +j q +‰ +}2 +ℓ2pGn`1q +ďR2 +n`1 +˜Kn`1 +ÿ +i“1 +Kn +ÿ +j“1 +}gn`1 +ij +pTn`1q}2 +op +¸ +}}f n ´ hn +j }2 +Ln +where the second to last step is an application of the Cauchy Schwarz inequality. +Proceeding inductively and using our previously established estimates, this proves the claim for all +settings in which Tn is nor normal (using an additional application of the triangle inequality for the +case of holomorphic filters). +To prove the claim for normal Tn as well, we note that in this setting we have (writing pφα, λαq|G| +α“1 +for a normalozed eigenvalue-eigenvector sequence of Tn`1) that we have +Kn`1 +ÿ +i“1 +››››› +Kn +ÿ +j“1 +gn`1 +ij +pTn`1q +“ +Pn`1pf n +j q ´ Pn`1phn +j q +‰ +››››› +2 +ℓ2pGn`1q +“ +Kn`1 +ÿ +i“1 +››››› +Kn +ÿ +j“1 +ÿ +α +gn`1 +ij +pλαqxφα, +“ +Pn`1pf n +j q ´ Pn`1phn +j q +‰ +yℓ2pGn`1qφα +››››› +2 +ℓ2pGn`1q +“ +Kn`1 +ÿ +i“1 +Kn +ÿ +j“1 +ÿ +α +|gn`1 +ij +pλαq|2|xφα, +“ +Pn`1pf n +j q ´ Pn`1phn +j q +‰ +yℓ2pGn`1q|2 +ď +ÿ +α +˜ÿ +i,j +|gijpλαq|2 +¸ Kn +ÿ +j“1 +|xφα, +“ +Pn`1pf n +j q ´ Pn`1phn +j q +‰ +yℓ2pGn`1q|2 +ď Bn`1Rn`1}}f n ´ hn +j }2 +Ln. +Here we applied Cauchy Schwarz once more in the second to last step and bounded +˜ÿ +i,j +|gijpλαq|2 +¸ +ď +˜ +sup +λPσpT q +ÿ +i,j +|gijpλq|2 +¸ +. +To see that these bounds are not necessarily tight, we may simply note that if we have a simple +one-layer Network as depicted in Fig. 9 below, the stability can be tightened to +}ΦNpfq ´ ΦNphq}Lout ď LRB ¨ }f ´ h}Lin +with with Bn :“ max +i“a,bpsupλPσpT q |gipλq|q as opposed to with Bn :“ +b +supλPσpT q +ř +i“a,b |gipλq|2 if +T is normal; as an easy calculation shows. +F +PROOF OF LEMMA 4.2 +We want to prove the following: +17 + +Figure 9: Sparsely connected Layer +Lemma F.1. Let T, rT be operators on on ℓ2pGq , ℓ2p rGq with }T}op, } rT}op ď C. +Let J : +ℓ2pGq Ñ ℓ2p rGq be arbitrary but linear. With Kg “ ř8 +k“1 |ag +k|kCk´1 for g entire and Kg “ +1 +2π +ű +BD +1 +zγT pzqγ rT pzq|gpzq|d|z| for g holomorphic, we have +}gpTqJ ´ Jgp rTq}op ď Kg ¨ }JT ´ rTJ}op +Proof. Let us first verify the claim for entire g. We first note that +rT kJ ´ JT k “ rT k´1p rTJ ´ JTq ` p rT k´1J ´ JT k´1qT +“ rT k´1p rTJ ´ JTq ` rT k´2p rTJ ´ JTqT ` p rT k´2J ´ JT k´2qT 2. +Thus, with }T}op, } rT}op ď C we find +} rT kJ ´ JT k}op ď kCk´1} rTJ ´ JT}op. +The claim now follows from applying the triangle inequality. +Now let us prove the bound for holomorphic g. We first note the following: +1 +rT ´ z +p rTJ ´ JTq +1 +T ´ z +“ +1 +rT ´ z +rTJ +1 +T ´ z ´ +1 +rT ´ z +JT +1 +T ´ z +“ +„ +1 +rT ´ z +p rT ´ zqJ ` +z +rT ´ z +ȷ +1 +T ´ z ´ +1 +rT ´ z +„ +1 +T ´ z pT ´ zqJ ` +z +T ´ z +ȷ +“z +ˆ +J +1 +T ´ z ´ +1 +rT ´ z +J +˙ +. +Thus we have +}gp rTqJ´JgpTq}op ď 1 +2π +¿ +BD +1 +|z|}RzpTq}op}Rzp rTq}op|gpzq|d|z| ď 1 +2π +¿ +BD +1 +|z|γT pzqγ rT pzq|gpzq|d|z|. +G +PROOF OF THEOREM 4.3 +We prove the following generalization of Theorem 4.3: +Theorem G.1. Let ΦN, rΦN be the maps associated to N-layer graph convolutional networks with +the same non-linearities and functional calculus filters, but based on different graph signal spaces +ℓ2pGq, ℓ2p rGq, characteristic operators Tn, rTn and connecting operators Pn, rPn. Assume Bn, rBn ď B +as well as Rn, rRn ď R and Ln ď L for some B, R, L ą 0 and all n ě 0. Assume that there are +identification operators Jn : ℓ2pGnq Ñ ℓ2p rGnq (0 ď n ď N) almost commuting with non- +linearities and connecting operators in the sense of } rPnJn´1f ´ JnPnf}ℓ2p r +Gnq ď δ2}f}ℓ2pGnq and +}ρnpJnfq´Jnρnpfq}ℓ2p r +Gnq ď δ1}f}ℓ2pGnq. Depending on whether normal or arbitrary characteristic +operators are used, define D2 +n :“ ř +jPKn´1 +ř +iPKn D2 +gn +ij or D2 +n :“ ř +jPKn´1 +ř +iPKn K2 +gn +ij. Choose +D such that Dn ď D for all n. Finally assume that }JnTn ´ rTnJn}˚ ď δ and with ˚ “ F if both +18 + +P→l2(Gout) +io +(Gout) +inooperators are normal and ˚ “ op otherwise. Then we have for all f P Lin and with JN the operator +that the KN copies of JN induced through concatenation that +}rΦpJ0fq ´ JNΦpfq} Ă +Lout ď N ¨ rRLDδ ` δ1BR ` δ2BLs ¨ pBRLqN´1 ¨ }f}Lin. +Proof. For simplicity in notation, let us denote the hidden representation of J0f in Ă +Ln by rf n. We +then note the following +}Jn`1f n`1 ´ rf n`1} Ă +Ln`1 +“ +¨ +˝ +Kn`1 +ÿ +i“1 +›››››Jn`1ρn`1 +˜ Kn +ÿ +j“1 +gn`1 +ij +pTn`1qPn`1pf n +j q +¸ +´ ρn`1 +˜ Kn +ÿ +j“1 +gn`1 +ij +pTn`1q rPn`1p rf n +j q +¸››››› +2 +ℓ2pGn`1q +˛ +‚ +1 +2 +ď +¨ +˝ +Kn`1 +ÿ +i“1 +›››››Jn`1ρn`1 +˜ Kn +ÿ +j“1 +gn`1 +ij +pTn`1qPn`1pf n +j q +¸ +´ ρn`1 +˜ +Jn`1 +Kn +ÿ +j“1 +gn`1 +ij +pTn`1qPn`1pf n +j q +¸››››› +2 +ℓ2pGn`1q +˛ +‚ +1 +2 +`L +¨ +˝ +Kn`1 +ÿ +i“1 +›››››Jn`1 +Kn +ÿ +j“1 +gn`1 +ij +pTn`1qPn`1pf n +j q ´ +Kn +ÿ +j“1 +gn`1 +ij +pTn`1q rPn`1p rf n +j q +››››› +2 +ℓ2pGn`1q +˛ +‚ +1 +2 +We can bound the first term by δ1B ¨ R ¨ pBRLqn ¨ }f}Lin. For the second term we find +L +¨ +˝ +Kn`1 +ÿ +i“1 +›››››Jn`1 +Kn +ÿ +j“1 +gn`1 +ij +pTn`1qPn`1pf n +j q ´ +Kn +ÿ +j“1 +gn`1 +ij +pTn`1q rPn`1p rf n +j q +››››› +2 +ℓ2pGn`1q +˛ +‚ +1 +2 +ďL +¨ +˝ +Kn`1 +ÿ +i“1 +››››› +Kn +ÿ +j“1 +pJn`1gn`1 +ij +pTn`1q ´ gn`1 +ij +p rTn`1qJn`1qPn`1pf n +j q +››››› +2 +ℓ2pGn`1q +˛ +‚ +1 +2 +`LB +˜ Kn +ÿ +j“1 +›››Jn`1Pn`1pf n +j q ´ rPn`1p rf n +j q +››› +2 +ℓ2pGn`1q +¸ 1 +2 +Arguing as in the proof of 3.1 we can bound the first term by LD ¨ δR ¨ pBRLqn}f}Lin. For the +second term we find, +LB +˜ Kn +ÿ +j“1 +›››Jn`1Pn`1pf n +j q ´ rPn`1p rf n +j q +››› +2 +ℓ2pGn`1q +¸ 1 +2 +ď LBδ2pBRLqn ` }Jnf n ´ rf n} Ă +Ln +arguing as above. Iterating from n “ N to n “ 0 then yields the claim. +H +TRANSFERABILITY: GENERAL CONSIDERATIONS +We first prove the statement made at the beginning of Section 5 that +}pωId ´ ∆δbq´1 ´ pωId ´ ∆δaq´1}op “ Opδa ¨ δbq. +To this end denote the increasing sequence of eigenvalues (counted without multiplicity) of ∆1 by +tλiuM +i“0. Recall that λ0 “ 0 Denote the sequence of projections on the corresponding eigenspaces by +tPiuM +i“0. We have for the resolvent that +1 +ωId ´ ∆δ +“ +1 +ωId ´ δ ¨ ∆1 +“ +M +ÿ +i“0 +1 +ω ´ 1 +δ λi +Pi. +19 + +Thus we have for δa, δb small enough that +›››› +1 +ωId ´ ∆δa +´ +1 +ωId ´ ∆δb +›››› +op +“ +ˇˇˇˇˇ +1 +ω ´ 1 +δa λ1 +´ +1 +ω ´ 1 +δb λ1 +ˇˇˇˇˇ “ +ˇˇˇˇˇλ1 +1 +δa ´ 1 +δb +pω ´ 1 +δa λ1qpω ´ 1 +δb λ1q +ˇˇˇˇˇ +“ λ1 +1 +|pω ´ 1 +δa λ1qpω ´ 1 +δb λ1q| “ Opδa ¨ δbq. +Next we note the convergence pωId´∆δq´1 Ñ P0 ¨pω ´0q´1. But this is obvious, since for λi ‰ 0 +we have +1 +ω ´ λi +δ +Ñ 0 +as δ Ñ 0. +I +PROOFS OF LEMMA 5.3 AND THEOREM 5.4 +Lemma I.1. Let T and rT be characteristic operators on ℓ2pGq and ℓ2p rGq be respectively. If these +operators are ω-δ-close with identification operator J, and }Rω}op, Rω}op ď C we have +}JgpTq ´ gp rTqJ}op ď Kg ¨ }p rRωJ ´ JRωq}op +with Kg “ +ű +BDp1 ` |z ´ ω|γT pzqqp1 ` |z ´ ω|γ rT pzqq|gpzq|d|z| if g is holomorphic and Kg “ +}g}F hol +ω,C if g P F hol +ω,C. If T and rT are normal as well as doubly ω-δ-close and g P F cont +ω,C , we have +Kg “ }g}F cont +ω,C . +Proof. We first deal with the statement concerning holomorphic g. To this end we note that Lemma +4.5.9 of Post (2012) proves +} rRzJ ´ JRz}op ď p1 ` |z ´ ω|γT pzqqp1 ` |z ´ ω|γ rT pzqq ¨ } rRωJ ´ JRω}op. +The claim then follows from +}JgpTq ´ gp rTqJ}op ď 1 +2π +¿ +BD +|gpzq|} rRzJ ´ JRz}opd|z|. +For g P F hol +ω,C the claim is proved exactly as in the proof of Lemma 2.3. +For g P F cont +ω,C we note that +p rRωqµp rR˚ +ωqνJ ´ J pRωqµ pR˚ +ωqν “ p rRωqµ ” +p rR˚ +ωqνJ ´ J pR˚ +ωqνı +` rp rRωqµJ ´ JpRωqµs pR˚ +ωqν . +Together with the result +} rT kJ ´ JT k}op ď kCk´1} rTJ ´ JT}op. +established in the proof of Lemma 4.2, the claim then follows from the triangle inequality together +with the definition of the semi-norm }g}F cont +ω,C . +As in the previous section, we state a slightly more general version of our main theorem of this +section: +Theorem I.2. Let Φ, rΦ be the maps associated to N-layer graph convolutional networks with +the same non-linearities and functional calculus filters, but based on different graph signal spaces +ℓ2pGnq, ℓ2p rGnq, characteristic operators Tn, rTn and connecting operators Pn, rPn. Assume Bn, rBn ď +B as well as Rn, rRn ď R and Ln ď L for some B, R, L ą 0 and all n ě 0. Assume that there +are identification operators Jn : ℓ2pGnq Ñ ℓ2p rGnq (0 ď n ď N) almost commuting with non- +linearities and connecting operators in the sense of } rPnJn´1f ´ JnPnf}ℓ2p r +Gnq ď δ2}f}ℓ2pGnq +and }ρnpJnfq ´ Jnρnpfq}ℓ2p r +Gnqδ1}f}ℓ2pGnq. define D2 +n :“ ř +jPKn´1 +ř +iPKn K2 +gn +ij with Kgn +ij as in +20 + +Lemma 5.3. Choose D such that Dn ď D for all n. Finally assume that }JnpωId ´ Tnq´1 ´ pωId ´ +rTnq´1Jn}op ď δ. If filters in F cont +ω,C are used, assume additionally that }JnppωId ´ Tnq´1q˚ ´ +ppωId ´ rTnq´1q˚Jn}op ď δ. Then we have for all f P Lin and with JN the operator that the KN +copies of JN induced through concatenation that +}rΦpJ0fq ´ JNΦpfq} Ă +Lout ď N ¨ rRLDδ ` δ1BR ` δ2BLs ¨ pBRLqN´1 ¨ }f}Lin. +Proof. The proof proceeds in complete analogy to the one of Theorem 4.3. +J +COLLAPSING STRONG EDGES: PROOFS AND FURTHER DETAILS +We utilize the notation introduced in Section 5.2. Beyond this, we denote the positive semi-definite +form induced by the energy functional E r +G by +E r +Gpu, vq :“ xu, ∆Gvyℓ2p r +Gq. +We further use the notation E r +Gpuq :“ E r +Gpu, uq. With +E r +G “ +ÿ +αP r +GGreek +βP r +GGreek +Ă +Wαβ|upαq ´ upβq|2 +` +ÿ +aP r +GLatin +bP r +GLatin +Ă +Wab|upaq ´ upbq|2 +` +ÿ +aP r +GLatin +βP r +GGreek +Ă +Waβ|upaq ´ upβq|2 +` +ÿ +αP r +GGreek +bP r +GLatin +Ă +Wαb|upαq ´ upbq|2 +` +ÿ +αP r +GGreek +Ă +Wα‹|upαq ´ up‹q|2 +` +ÿ +βP r +GGreek +Ă +W‹β|up‹q ´ upβq|2 +` +ÿ +aP r +GLatin +Ă +Wa‹|upaq ´ up‹q|2 +` +ÿ +bP r +GLatin +Ă +W‹b|up‹q ´ upbq|2 +(9) +Similar considerations apply when rG is replaced by G. +Let us next solve the convex optimization program (5) introduced in Definition 5.5, restated here for +convenience: +Definition J.1. For each g P G, define the signal ψδ +g P ℓ2p rGq as the unique solution to the convex +optimization program +min E r +Gpuq subject to uphq “ δhg for all h P rGLatin +ď +t‹u. +As a first step we note that all entries of ψg are real and non-negative, which follows since each +summand in (9) is non-increasing under the map u ÞÑ |u| due to the reverse triangle ||a|´|b|| ď |a´b|. +To find the explicit form of ψg, fix g P rGLatin +Ťt‹u and denote by χg P ℓ2p rGq the signal defined by +setting it to χηphq “ δhg for h P rGLatin +Ťt‹u and ηgpαq “ ηα +g with tηα +g uαP r +GGreek a set of | rGGreek| +21 + +free parameters in Rď0. We then have +E r +Gpχgq “2 +ÿ +aP r +GLatin +Ă +Wag ` 2 +ÿ +αP r +GGreek +Ă +Wαg|1 ´ ηα +g |2 ` 2 +ÿ +αP r +GGreek +bP r +GLatin +Ťt‹u +Ă +Wαb|ηα +g |2 +` +ÿ +α,βP r +GGreek +Ă +Wαβ|ηα +g ´ ηβ +g |2. +By definition, χg depends smoothly on the parameters tηα +g uαP r +GGreek. Finding the minimizer of the +convex optimization program (5) is then equivalent to finding the values tηα +g uαP r +GGreek at which we +have +BE r +Gpχgq +Bηαg +“ 0. +We note +1 +4 +BE r +Gpχgq +Bηξ +g +“ +¨ +˚ +˚ +˝Ă +Wgξ ` +ÿ +aP r +GLatin +a‰g +Ťt‹u +Ă +Wgξ ` +ÿ +αP r +GGreek +Ă +Wαg +˛ +‹‹‚ηg +ξ ´ +ÿ +αP r +GGreek +Ă +Wαgηg +α ´ Ă +Wgξ +Collecting these equations for all parameters into a matrix equation, we find that the ’Greek entries’ +of the vector ψg are given explicitly by +¨ +˚ +˝ +ψgpαq +ψgpβq +... +˛ +‹‚“ +¨ +˚ +˚ +˝ +rdα +´Ă +Wαβ +. . . +´Ă +Wβα +rdβ +... +... +. . . +... +˛ +‹‹‚ +´1 +¨ +¨ +˚ +˝ +Ă +Wgα +Ă +Wgβ +... +˛ +‹‚, +(10) +with degrees in rG denoted by rdα. Let us denote the restriction of ψδ +g to Greek entries, thought of as a +vector in C| r +GGreek| by ⃗ηδ +g. +Given the degree rdα corresponding to a Greek index, we decompose it as +rdα “ rdr +α ` Ă +Wα‹ ` Vα +with rdr +α accounting for edges from α to other greek vertices +rdr +α “ +ÿ +βP r +GGreek +Ă +Wαβ “ 1 +δ +ÿ +βP r +GGreek +ωαβ, +and Vα accounting for edges from α to Latin vertices +Vα “ +ÿ +aP r +GLatin +Ă +Waα. +Recall that we also may write +Ă +Wα‹ “ 1 +δ ωα‹. +We may then write +¨ +˚ +˚ +˝ +rdα +´Ă +Wαβ +. . . +´Ă +Wβα +rdβ +... +... +. . . +... +˛ +‹‹‚“ +¨ +˚ +˚ +˝ +rdr +α +´Ă +Wαβ +. . . +´Ă +Wβα +rdr +β +... +... +. . . +... +˛ +‹‹‚` 1 +δ +¨ +˚ +˚ +˝ +ωα‹ +0 +. . . +0 +ωβ‹ +... +... +. . . +... +˛ +‹‹‚` +¨ +˚ +˚ +˝ +Vα +0 +. . . +0 +Vβ +... +... +. . . +... +˛ +‹‹‚ +“: 1 +δ L ` 1 +δ diagp⃗ω‹q ` V, +where we made the obvious definitions for the matrices L and V and denoted by ⃗ω‹ the vector with +entries ωα‹. Let us also use the notation +h :“ L ` diagpω‹q. +22 + +Next we want to establish that h is invertible. For this we first note that that L is the graph Laplacian +of the subgraph rGGreek; which we assume to be connected. Hence L is positive semi-definite with +the eigenspace corresponding to the eigenvalue zero being spanned by (entry-wise) constant vectors. +Since all entries of ω‹ are non-negative, the operator h is also positive semi-definite. Since we assume +that the vertex ‹ is connected to at least one other vertex in rGGreek, there is at least one entry in +⃗ω‹ that is strictly greater than zero. We show that this already implies that h is in fact also positive +definite and hence invertible. Indeed, for any ⃗v P C| r +GGreek| we have +x⃗v, L ¨ ⃗vyC|Ă +GGreek| “ x⃗v, h ¨ ⃗vyC|Ă +GGreek| ` x⃗v, diagp⃗ω‹q ¨ ⃗vyC|Ă +GGreek|. +Both terms on the right hand side are non-negative. If ⃗v is a constant (non-zero) vector, the first term +vanishes, but since at least one entry of ω‹ is strictly positive, with all others being non-negative, the +second term on the right hand side is strictly positive. If ⃗v is non-constant, the first term on the right +hand side is larger than zero. Hence h is positive definite and thus invertible. Similarly one proves +that (for any δ ě 0) the operator h ` δV is positive definite and hence invertible. Thus we now know +that the operator +1 +δ ph ` δV q “ +¨ +˚ +˚ +˝ +rdα +´Ă +Wαβ +. . . +´Ă +Wβα +rdβ +... +... +. . . +... +˛ +‹‹‚ +utilized in (10) is indeed invertible. We note (again with the restriction of ψδ +g to Greek entries thought +of as a vector in C| r +GGreek| denoted by ⃗ηδ +g) that we may equivalently write (10) as +ph ` δV q´1⃗ηδ +g “ δ ⃗Ă +Wg +(11) +and +⃗Ă +Wg :“ +¨ +˚ +˝ +Ă +Wgα +Ă +Wgβ +... +˛ +‹‚ +thought of as an element of C| r +GGreek|. To proceed, we now first focus on the case g “ ‹, for which +we may write (11) equivalently as +ph ` δV q´1⃗ηδ +‹ “ ⃗ω‹. +(12) +Since ⃗ω‹ is independent of δ, we may take the limit δ Ñ 0 and arrive at +pL ` diagp⃗ω‹qq⃗η0 +‹ “ ⃗ω‹ +which is uniquely solved by ⃗η0 +‹ “ p1, 1, 1, ....q ” 1Greek. +Since we assume δ ! 1, we can now investigate the solution ⃗ηδ +g for non-zero δ through perturbation +theory. We write +⃗ηδ +‹ “ 1 r +GGreek ´ ⃗ζδ +‹ +with ⃗ζ0 +‹ “ 0 and find from (12) – using h ¨ 1Greek “ ⃗ηδ +‹ – the defining equation +⃗ζδ +‹ “ δph ` δV q´1 ¨ V ¨ 1 r +GGreek. +From this we obtain the estimate +}⃗ζδ +‹}ℓ2p r +GGreekq ď }ph ` δV q}op ¨ }V ¨ 1 r +GGreek}ℓ2p r +GGreekq ¨ δ, +where we denote by ℓ2p rGGreekq the space graph signal space C| r +GGreek| equipped with node weights +trµgugP r +GGreek. +We note that both h and V are positive semi-definite and we thus obtain +λminphq ď λminph ` δV q +23 + +for the minimal eigenvalues of the respective operators. Hence +}ph ` δV q´1}op ď }h´1}op, +and thus also +}⃗ζδ +‹}ℓ2p r +GGreekq ď }h´1}op ¨ }V ¨ 1 r +GGreek}ℓ2p r +GGreekq +looooooooooooooooomooooooooooooooooon +“:K +¨δ. +(13) +Since }h´1}op “ 1{λminphq we may write +K “ +}V ¨ 1Greek}ℓ2p r +GGreekq +λminphq +. +(14) +From (11) we know that for g ‰ ‹ we have ⃗ηδ +g “ 0. +We now also want to bound }⃗ηδ +g}ℓ2p r +GGreekq in terms of δ. We will do this by establishing the relationship +ÿ +gP r +GLatin +⃗ηδ +g “ ⃗ζδ +‹. +(15) +and then utilizing our estimate on }⃗ζδ +‹}ℓ2p r +GGreekq established above. To prove (15), we will need the +concept of harmonic extensions: +Definition J.2. Denote by ℓ2p rGLatin Y t‹uq the graph signal space C| r +GLatinYt‹u| equipped with the +node weights trµgugP r +GLatinYt‹u. Given an arbitrary signal u P ℓ2p rGLatin Y t‹uq a harmonic extension +of u to all of ℓ2p rGq is a signal u P ℓ2p rGq satisfying +p∆ r +Guqpαq “ 0 @α P rGGreek and uphq “ uphq @ h P rGLatin +ď +t‹u. +We first note that the concept of harmonic extensions is both well-defined an well-behaved: +Lemma J.3. Fix u P ℓ2p rGLatin Y t‹uq. There exists a unique harmonic extension u P ℓ2p rGq of u. +It is given as the solution to the convex optimization program +min E r +Gpuq subject to uphq “ δhg for all h P rGLatin +ď +t‹u. +Furthermore if u and v are the harmonic extensions of u and v, then pu ` vq is the (unique) harmonic +extension of pu ` vq. +Proof. We write a signal ψ P ℓ2p rGq as ψ “ pψ, ηq with ψ P ℓ2p rGLatin Y t‹uq and η P ℓ2p rGGreekq. +We then notice +ψ “ argminE r +Gpuq subject to ψphq “ ψphq for all h P rGLatin +ď +t‹u +ôBE r +Gpψq +Bηα +“ 0 @α P rGGreek and ψphq “ ψphq for all h P rGLatin +ď +t‹u +ô +ÿ +yP r +G +Ă +Wαypψpαq ´ ψpyqq “ 0 @α P rGGreek and ψphq “ ψphq for all h P rGLatin +ď +t‹u +ôp∆ r +Gψqpαq “ 0 @α P rGGreek and ψphq “ ψphq for all h P rGLatin +ď +t‹u. +Here, we treated ηα and its complex conjugate as independent variables and used that E r +Gp¨q is +a real-valued functional for the first equivalence. As harmonic extensions are thus equivalently +characterised as the solutions of convex minimization programs, they are unique. +To prove the last statement, we note that by linearity of the graph Laplacian, pu ` vq certainly is a +harmonic extension of pu ` vq. Since harmonic extensions are unique, it is the only one. +After this preparatory effort, we are now ready to prove (15): +24 + +Lemma J.4. For any δ ě 0 the signals t⃗ηδ +gugP r +GLatin +Ťt‹u form a partition of unity of ℓ2p rGGreekq: +ÿ +gP r +GLatin +Ťt‹u +⃗ηδ +g “ 1 r +GGreek +(16) +Equivalently we have +ÿ +gP r +GLatin +⃗ηδ +g “ ⃗ζδ +‹. +As an immediate Corollary we obtain +Corollary J.5. For any δ ě 0 the signals tψδ +gugP r +GLatin +Ťt‹u form a partition of unity of ℓ2p rGq: +ÿ +gP r +GLatin +Ťt‹u +⃗ηδ +g “ 1 r +G. +(17) +Proof. Using the ’boundary conditions’ in (5), it is straightforward to verify that (16) is equivalent to +(17). From Lemma J.3 we now know that ψδ +g, originally characterised as the solution of the problem +min E r +Gpuq subject to uphq “ δhg for all h P rGLatin +ď +t‹u, +is equivalently characterised as the harmonic extension of uphq “ δhg. From the last statement of +Lemma J.3, we know that ř +gP r +GLatin +Ťt‹u ⃗ηδ +g is the unique harmonic extension of +ÿ +gP r +GLatin +Ťt‹u +δhg “ 1 r +GĂ +GLatin +Ťt‹u. +But this – in turn – is the unique solution of the problem +min E r +Gpuq subject to uphq “ 1 for all h P rGLatin +ď +t‹u. +Since we have +E r +Gp1 r +Gq “ 0, +which is the lowest possible attainable value of E r +Gp¨q, and setting u “ 1 r +G is compatible with the +’boundary condition’ uphq “ 1 for all h P rGLatin +Ťt‹u, we know that is the (unique) harmonic +extension of 1 r +GLatin +Ťt‹u. By the last statement of Lemma J.3 we thus have +ÿ +gP r +GLatin +Ťt‹u +⃗ηδ +g “ 1 r +G. +Having established that we may write +ÿ +gP r +GLatin +⃗ηδ +g “ ⃗ζδ +‹, +together with the fact that every entry of each ⃗ηδ +g is non-negative, we now know that +0 ď ⃗ηδ +gpαq, ⃗ζδ +‹ ď 1. +Furthermore – using our earlier estimate (13) – we now easily obtain +›››››› +ÿ +gP r +GLatin +⃗ηδ +g +›››››› +ℓ2p r +GGreekq +ď K ¨ δ. +Hence – by positivity of the entries – we also have for each individual g P rGLatin that +››⃗ηδ +g +›› +ℓ2p r +GGreekq ď K ¨ δ. +25 + +For the weights tµδ +gugPG we then find +rµg ď µδ +g ď rµg ` δK +ÿ +αP r +GGreek +rµα +if g ‰ ‹. We also write rµp rGGreekq :“ ř +αP r +GGreek rµα. If g “ ‹, we have +rµδ +‹ ` p1 ´ δqrµp rGGreekq ď µδ +‹ ď rµδ +‹ ` rµp rGGreekq. +Having set the scene, we are now ready to prove Theorem 5.4. Following Post & Simmer (2017), +instead of checking the conditions of Definition 5.1 and Definition 5.2 it is instead sufficient to check +the following, with J rJ as defined in Section 5.2 to establish Theorem 5.6: +Lemma J.6. In addition to identification operators J, rJ, assume that there exist additional operators +J1 : ℓ2pGq Ñ ℓ2p rGq and rJ1 : ℓ2p rGq Ñ ℓ2pGq so that the following set of equations is satisfied with +ϵ “ Opδ +1 +2 q +}Jf} ď p1 ` ϵ1q}f}, +|xJf, uy ´ xf, rJuy| ď ϵ1}f} +(18) +}f ´ rJJf} ď ϵ1a +}f}2 ` EGpfq, +}u ´ J rJu} ď ϵ1b +}u}2 ` E r +Gpuq +(19) +}J1f ´ Jf} ď ϵ1a +}f}2 ` EGpfq, +} rJu ´ rJ1u} ď ϵ1b +}u}2 ` E r +Gpuq +(20) +}E r +GpJ1f, uq ´ EGpf, rJ1uq} ď ϵ1 ¨ +a +}f}2 ` EGpfq ¨ +b +}u}2 ` E r +Gpuq. +(21) +Then the (normal) operators ∆ and r∆ are (doubly) (-1)- (ϵ “ 12ϵ1) -close with identification-operator +J. +Here, we always have u P ℓ2p rGq and f P ℓ2pGq) +Proof. This follows immediately after combining Proposition 4.4.12 with Theorem 4.4.15 of Post +(2012). +We set J1f “ Jf and p rJ1uqpxq “ upxq and now determine the individual ϵ “ ϵpδq values for which +these equations are satisfied: +Left-hand-side of (18): +For the left hand side of (18) we note (using 2ab ď a2 ` b2 and the fact that the ψg form a partition +of unity): +}Jf}2 +ℓ2p r +Gq “ +ÿ +h,gPG +xψδ +h, ψδ +gyℓ2p r +Gqfphqfpgq +ď 1 +2 +ÿ +hPG +|fphq|2 ÿ +gPG +xψδ +h, ψgyℓ2p r +Gq ` 1 +2 +ÿ +gPG +|fpgq|2 ÿ +hPG +xψδ +h, ψδ +gyℓ2p r +Gq +“ 1 +2 +ÿ +hPG +|fphq|2xψδ +h, 1yℓ2p r +Gq ` 1 +2 +ÿ +gPG +|fpgq|2x1, ψδ +gyℓ2p r +Gq +“ +ÿ +gPG +|fpgq|2µδ +g +“ }f}2 +ℓ2pGq. +Here the second to last inequality follows from the definition of the weights µδ +g. Thus the left hand +side of (18) holds with +ϵ “ 0. +26 + +Right-hand-side of (18): +The right hand side of (18) holds trivially with +ϵ “ 0 +since we have chosen J˚ “ rJ. +Left-hand-side of (19): +Now let us check the l.h.s. of (19). We have: +pf ´ rJJfqpyq “ fpyq ´ +ÿ +gPG +fpgq +xψδ +g, ψδ +yyℓ2p r +Gq +µδy +. +Using the constant K defined in (14) we have +rµg ď µδ +g ď rµg ` δK +ÿ +αP r +GGreek +rµα +if g ‰ ‹. We also write rµp rGGreekq :“ +ř +αP r +GGreek +rµα. If G “ ‹, we have +rµ‹ ` p1 ´ δqrµp rGGreekq ď µδ +‹ ď rµ‹ ` 1rµp rGGreekq. +We next note +xψδ +x, ψδ +yyℓ2p r +Gq “ rµxδxy ` x⃗ηδ +x, ⃗ηδ +yyℓ2pGGreekq +with Ă +Wx the vector with entries Ă +Wxpgq “ Ă +Wxg. +Thus for y ‰ ‹ we find +|pf ´ rJJfqpyq| ď +ˆ +1 ´ rµy +µδy +˙ +|fpyq| ` +ˇˇˇˇˇˇˇ +ÿ +gPG +g‰y +fpgq +xψδ +g, ψδ +yyℓ2p r +Gq +µδy +ˇˇˇˇˇˇˇ +. +We thus find +}f ´ rJJf}ℓ2pGq ď +g +f +f +f +f +e +ÿ +yPG +y‰‹ +¨ +˚ +˝ +ˆ +1 ´ rµy +µδy +˙ +|fpyq| ` +ˇˇˇˇˇˇˇ +ÿ +gPG +g‰y +fpgq +xψδg, ψδyyℓ2p r +Gq +µδy +ˇˇˇˇˇˇˇ +˛ +‹‚ +2 +` +ˇˇˇˇˇfp‹q ´ +ÿ +gPG +fpgq +xψδ +g, ψδ +‹yℓ2p r +Gq +µδ‹ +ˇˇˇˇˇ +ď +g +f +f +f +e +ÿ +yPG +y‰‹ +ˆˆ +1 ´ rµy +µδy +˙ +|fpyq| +˙2 +` +g +f +f +f +f +e +ÿ +yPG +y‰‹ +¨ +˚ +˝ +ˇˇˇˇˇˇˇ +ÿ +gPG +g‰y +fpgq +xψδg, ψδyyℓ2p r +Gq +µδy +ˇˇˇˇˇˇˇ +˛ +‹‚ +2 +` +ˇˇˇˇˇfp‹q ´ +ÿ +gPG +fpgq +xψδ +g, ψδ +‹yℓ2p r +Gq +µδ‹ +ˇˇˇˇˇ +To bound the first term of the estimate, we note (for y ‰ ‹) and δ small enough: +ˆ +1 ´ rµy +µδy +˙ +ď +˜ +1 ´ +rµy +rµy ` δKrµp rGGreekq +¸ +“ +δKrµp rGGreekq +δKrµy ` rµp rGGreekq +ď δ Krµp rGGreekq +min +gP r +GLatin +rµg +. +27 + +We also note (for y ‰ ‹) +|fpyq| ď +1 +min +gP r +GLatin +?µg +|fpyq|?µy ď +1 +min +gP r +GLatin +a +rµy +|fpyq|?µy +Thus we find +g +f +f +f +e +ÿ +yPG +y‰‹ +ˆˆ +1 ´ rµy +µδy +˙ +|fpyq| +˙2 +ď δ +¨ +˚ +˚ +˝ +Krµp rGGreekq +min +gP r +GLatin +rµ +3 +2g +˛ +‹‹‚ +g +f +f +e +ÿ +yPG +y‰‹ +|fpyq|2µy ď δ +¨ +˚ +˚ +˝ +Krµp rGGreekq +min +gP r +GLatin +rµ +3 +2g +˛ +‹‹‚}f}ℓ2pGq. +To estimate the second term, we estimate +|fpgq| ď +1 +min +gP r +GLatinYt‹u +a +rµg +}f}ℓ2pGq +to obtain +ˇˇˇˇˇˇˇ +ÿ +gPG +g‰y +fpgq +xψδ +g, ψδ +yyℓ2p r +Gq +µδy +ˇˇˇˇˇˇˇ +ď +¨ +˚ +˝ +1 +min +gP r +GLatinYt‹u +a +rµg +˛ +‹‚}fpyq}ℓ2pGq ¨ +ˇˇˇˇˇˇˇ +ÿ +gPG +g‰y +xψδ +g, ψδ +yyℓ2p r +Gq +µδy +ˇˇˇˇˇˇˇ +“ +¨ +˚ +˝ +1 +min +gPGLatinYt‹u +a +rµg +˛ +‹‚}fpyq}ℓ2pGq ¨ +ˇˇˇˇˇˇˇ +ÿ +gPG +g‰y +x⃗ηδ +g, ⃗ηδ +yyℓ2p r +GGreekq +µδy +ˇˇˇˇˇˇˇ +ď +¨ +˚ +˝ +1 +min +gP r +GLatinYt‹u +a +rµg +˛ +‹‚}fpyq}ℓ2pGq ¨ +ˇˇˇˇˇˇˇ +ÿ +gPG +g‰y +x⃗ηδ +g, ⃗ηδ +yyℓ2p r +GGreekq +rµy +ˇˇˇˇˇˇˇ +Thus we find (using that x⃗ηδ +g, ⃗ηδ +yyℓ2p r +GGreekq is a non-negative number and we have } ¨ }2 ď } ¨ }1) +g +f +f +f +f +e +ÿ +yPG +y‰‹ +¨ +˚ +˝ +ˇˇˇˇˇˇˇ +ÿ +gPG +g‰y +fpgq +xψδg, ψδyyℓ2p r +Gq +µδy +ˇˇˇˇˇˇˇ +˛ +‹‚ +2 +ď +1 +min +gP r +GLatinYt‹u +a +rµg +}f}ℓ2pGq ¨ +ÿ +yPG +y‰‹ +ÿ +gPG +g‰y +x⃗ηδ +g, ⃗ηδ +yyℓ2p r +GGreekq +rµy +ď +1 +min +gP r +GLatinYt‹u +rµ +3 +2g +}f}ℓ2pGq ¨ +ÿ +yPG +y‰‹ +ÿ +gPG +g‰y +x⃗ηδ +g, ⃗ηδ +yyℓ2p r +GGreekq +ď +1 +min +gP r +GLatinYt‹u +rµ +3 +2g +}f}ℓ2pGq ¨ +ÿ +yPG +y‰‹ +ÿ +gPG +x⃗ηδ +g, ⃗ηδ +yyℓ2p r +GGreekq +ď +1 +min +gP r +GLatinYt‹u +rµ +3 +2g +}f}ℓ2pGq ¨ x1 r +GGreek, ⃗ζδ +‹yℓ2p r +GGreekq +ď +1 +min +gP r +GLatinYt‹u +rµ +3 +2g +}f}ℓ2pGq ¨ }1 r +GGreek}ℓ2p r +GGreekq ¨ }⃗ζδ +‹}ℓ2p r +GGreekq +ď δ ¨ +¨ +˚ +˚ +˝ +K ¨ +b +rµp rGGreekq +min +gP r +GLatinYt‹u +rµ +3 +2g +˛ +‹‹‚}f}ℓ2pGq +Let us thus turn to the remaining term; corresponding to y “ ‹: We have +28 + +ˇˇˇˇˇfp‹q ´ +ÿ +gPG +fpgq +xψδ +g, ψδ +‹yℓ2p r +Gq +µδ‹ +ˇˇˇˇˇ ď +ˇˇˇˇˇ1 ´ +xψδ +‹, ψδ +‹yℓ2p r +Gq +µδ‹ +ˇˇˇˇˇ |fp‹q| ` +ˇˇˇˇˇˇˇ +ÿ +gPG +g‰‹ +fpgq +xψδ +g, ψδ +‹yℓ2p r +Gq +µδ‹ +ˇˇˇˇˇˇˇ +(22) +We first deal with the left summand. We note +ˇˇˇˇˇ1 ´ +xψδ +‹, ψδ +‹yℓ2p r +Gq +µδ‹ +ˇˇˇˇˇ “ +ˇˇˇˇˇ +µδ +‹ ´ rµ‹ ´ x1 r +GGreek ´ ⃗ζδ +‹, 1 r +GGreek ´ ⃗ζδ +‹yℓ2p r +GGreekq +µδ‹ +ˇˇˇˇˇ +ď +ˇˇˇˇˇ +µδ +‹ ´ rµ‹ ´ x1 r +GGreek ´ ⃗ζδ +‹, 1 r +GGreek ´ ⃗ζδ +‹yℓ2p r +GGreekq +rµ‹ ` rµp rGGreekq ´ δKrµp rGGreekq +ˇˇˇˇˇ +ď +ˇˇˇˇˇˇ +´ +µδ +‹ ´ rµ‹ ´ x1 r +GGreek, 1 r +GGreekyℓ2p r +GGreekq +¯ +` +´ +x⃗ζδ +‹, ⃗ζδ +‹yℓ2p r +GGreekq ´ 2x1 r +GGreek, ⃗ζδ +‹yℓ2p r +GGreekq +¯ +rµ‹ ` rµp rGGreekq ´ δKrµp rGGreekq +ˇˇˇˇˇˇ +ď +pδKq ` +ˇˇˇx⃗ζδ +‹, ⃗ζδ +‹yℓ2p r +GGreekq ´ 2x1 r +GGreek, ⃗ζδ +‹yℓ2p r +GGreekq +ˇˇˇ +rµ‹ ` rµp rGGreekq ´ δKrµp rGGreekq +ď +pδKq ` δ2K2 ` 2}1 r +GGreek}ℓ2p r +GGreekq ¨ }⃗ζδ +‹}ℓ2p r +GGreekq +rµ‹ ` rµp rGGreekq ´ δKrµp rGGreekq +ď +pδKq ` +ˇˇˇx⃗ζδ +‹, ⃗ζδ +‹yℓ2p r +GGreekq ´ 2x1 r +GGreek, ⃗ζδ +‹yℓ2p r +GGreekq +ˇˇˇ +rµ‹ ` rµp rGGreekq ´ δKrµp rGGreekq +ď +pδKq ` δ2K2 ` 2 +b +rµp rGGreekqKδ +rµ‹ ` rµp rGGreekq ´ δKrµp rGGreekq +ď +pδKq ` δ2K2 ` 2 +b +rµp rGGreekqKδ +rµ‹ +Thus, under the assumption δ ď 1 (implying δ2 ď δ), we have +ˇˇˇˇˇ1 ´ +xψδ +‹, ψδ +‹yℓ2p r +Gq +µδ‹ +ˇˇˇˇˇ ď +K ` K2 ` 2 +b +rµp rGGreekqK +rµ‹ +¨ δ. +This implies that we have +ˇˇˇˇˇfp‹q ´ +ÿ +gPG +fpgq +xψδ +g, ψδ +‹yℓ2p r +Gq +µδ‹ +ˇˇˇˇˇ ď δ ¨ +K ` K2 ` 2 +b +rµp rGGreekqK +rµ +3 +2‹ +¨ }f}ℓ2pGq. +29 + +For the right-hand-side summand of the estimate in (22) we note +ˇˇˇˇˇˇˇ +ÿ +gPG +g‰‹ +fpgq +xψδ +g, ψδ +‹yℓ2p r +Gq +µδ‹ +ˇˇˇˇˇˇˇ +“ +ˇˇˇˇˇˇˇ +ÿ +gPG +g‰‹ +fpgq +x⃗ηδ +g, ⃗ηδ +‹yℓ2p r +GGreekq +µδ‹ +ˇˇˇˇˇˇˇ +ď +1 +min +gP r +GLatinYt‹u +rµ +3 +2g +}f}ℓ2pGq +ÿ +gPG +g‰‹ +x⃗ηδ +g, ⃗ηδ +‹yℓ2p r +GGreekq +ď +1 +min +gP r +GLatinYt‹u +rµ +3 +2g +}f}ℓ2pGq +ÿ +gPG +x⃗ηδ +g, ⃗ηδ +‹yℓ2p r +GGreekq +“ +1 +min +gP r +GLatinYt‹u +rµ +3 +2g +}f}ℓ2pGqx1 r +GGreek, ⃗ζδ +‹yℓ2p r +GGreekq +δ ¨ +¨ +˚ +˚ +˝ +K ¨ +b +rµp rGGreekq +min +gP r +GLatinYt‹u +rµ +3 +2g +˛ +‹‹‚}f}ℓ2pGq. +Putting it all together, we find for δ ď 1 that +}f ´ rJJf}ℓ2pGq ď δ ¨ KA ¨ }f}ℓ2pGq +with +KA :“ +¨ +˚ +˚ +˝ +Krµp rGGreekq +min +gP r +GLatin +rµ +3 +2g +˛ +‹‹‚` 2 +¨ +˚ +˚ +˝ +K ¨ +b +rµp rGGreekq +min +gP r +GLatinYt‹u +rµ +3 +2g +˛ +‹‹‚` +K ` K2 ` 2 +b +rµp rGGreekqK +rµ +3 +2‹ +. +Thus the left hand side of (19) holds with +ϵ “ KA ¨ δ. +Right-hand-side of (19): +Hence let us now check the right hand side of (19). We note +pu ´ J rJuq “ u ´ +ÿ +xPG +xψδ +x, uyℓ2p r +Gq +µδx +ψδ +x. +Let us denote by M the matrix representation +M δ “ Id ´ rJJ “ Id ´ +ÿ +xPG +xψδ +x, ¨yℓ2p r +Gq +µδx +ψδ +x. +We use the triangle inequality to arrive at +›››pu ´ J rJuq +››› +ℓ2p r +Gq ď +››M 0 ¨ u +›› +ℓ2p r +Gq ` +››M δ ´ M 0›› +op ¨ }u}ℓ2p r +Gq . +(23) +Using the fact that for g ‰ ‹ we have ⃗ηδ +g Ñ ⃗0 an ⃗η0 +‹ “ 1 r +GGreek we find in the (δ Ñ 0)-limit that +M 0 “ +˜ +0| r +GLatin|ˆ| r +GLatin| +0| r +GLatin|ˆ| r +GGreekYt‹u| +0| r +GGreekYt‹u|ˆ| r +GLatin| +M 0 +¸ +with +30 + +M 0 “ +¨ +˚ +˝ +1 +... +1 +˛ +‹‚´ +1 +rµp rGGreekq ` rµ‹ +¨ +˚ +˝ +rµ‹ +rµα +rµβ +¨ ¨ ¨ +rµ‹ +rµα +rµβ +¨ ¨ ¨ +... +... +... +˛ +‹‚ +acting on ℓ2p rGGreek Y t‹uq . For any element v P ℓ2p rGq, let us denote its restriction to rGGreek Y t‹uby +v P ℓ2p rGGreek Y t‹uq . +We thus find +››M 0 ¨ u +››2 +ℓ2p r +GGreekYt‹u “ xM 0 ¨ u, M 0 ¨ uyℓ2p r +GGreekYt‹u +“ +ÿ +iP r +GGreekYt‹u +ÿ +jP r +GGreekYt‹u +upiqupjq +ÿ +a,bP r +GGreekYt‹u +« +δia ´ +rµi +rµp rGGreekq ` rµ‹ +ff +¨ rµaδab ¨ +« +δbj ´ +rµj +rµp rGGreekq ` rµ‹ +ff +“ +ÿ +iP r +GGreekYt‹u +ÿ +jP r +GGreekYt‹u +upiqupjq +ÿ +aP r +GGreekYt‹u +« +δia ´ +rµi +rµp rGGreekq ` rµ‹ +ff +¨ +« +rµaδaj ´ +rµarµj +rµp rGGreekq ` rµ‹ +ff +“ +ÿ +iP r +GGreekYt‹u +ÿ +jP r +GGreekYt‹u +upiqupjq ˆ ... +... ˆ +ÿ +aP r +GGreekYt‹u +« +rµarµ‹δiaδaj ´ +δiarµarµj +rµp rGGreekq ` rµ‹ +´ +δijrµirµj +rµp rGGreekq ` rµ‹ +` +rµirµarµj +prµp rGGreekq ` rµ‹q2 +ff +“ +ÿ +iP r +GGreekYt‹u +ÿ +jP r +GGreekYt‹u +upiqupjq +« +rµiδij ´ +rµirµj +rµp rGGreekq ` rµ‹ +ff +“ +ÿ +i,jP r +GGreekYt‹u +˜ +rµirµj +rµp rGGreekq ` rµ‹ +¸ +|upiq ´ upjq|2. +To proceed, we prove the following Lemma: +Lemma J.7. Let i, j P rGGreek Y t‹u. Denote by C r +GGreekYt‹upi, jq the minimum number of edges for +which ωij ŋ 0 needed to connect i and j by a path. Set +C r +GGreekYt‹u :“ +max +i‰jP r +GGreekYt‹u +C r +GGreekYt‹upi, jq. +Furthermore set +Ω :“ +min +i‰jP r +GGreekYt‹u +ωij. +We have +|upiq ´ upjq| ď δ +1 +2 +˜ +C r +GGreekYt‹u +? +Ω +¸ b +E r +Gpuq. +We call C r +GGreekYt‹u the connectivity constant of the sub-graph rGGreek Y t‹u and note that it is +well-defined since we assume rGGreek Y t‹u to be connected. +31 + +Proof. Fix i and j. Let ti, g1, ..., gn, ju be the vertices traversed by a path of minimal length +determining C r +GGreekYt‹upi, jq. We then have +|upiq ´ upjq| +ď|upiq ´ upg1q| ` |upg1q ´ upg2q| ` ... ` |upgnq ´ upjq| +ďδ +1 +2 +1 +? +Ω +ˆb +Ă +Wig1|upiq ´ upg1q|2 ` +b +Ă +Wg1g2|upg1q ´ upg2q|2 ` ... ` +b +Ă +Wgnj||upgnq ´ upjq|2 +˙ +ďδ +1 +2 +1 +? +Ω +´b +E r +Gpuq ` +b +E r +Gpuq ` ... ` +b +E r +Gpuq +¯ +“δ +1 +2 C r +GGreekYt‹upi, jq +? +Ω +b +E r +Gpuq +ďδ +1 +2 C r +GGreekYt‹u +? +Ω +b +E r +Gpuq. +With the help of this Lemma we then find +››M 0 ¨ u +›› +ℓ2p r +Gq ď δ +1 +2 C r +GGreekYt‹u +? +Ω +b +E r +Gpuq ¨ +g +f +f +e +ÿ +i,jP r +GGreekYt‹u +˜ +rµirµj +rµp rGGreekq ` rµ‹ +¸ +“ δ +1 +2 ¨ +¨ +˝C r +GGreekYt‹u ¨ +b +rµp rGGreekq ` rµ‹ +? +Ω +˛ +‚¨ +b +E r +Gpuq. +To derive a bound for +››M δ ´ M 0›› +op in the second term of the estimate (23), we write +M δ ´ M 0 “ +ˆ +B +A +A: +D +˙ +. +Here we denote by +A: : ℓ2p rGLatinq ÝÑ ℓ2p rGGreek Y t‹uq +the adjoint of the operator +A : ℓ2p rGGreek Y t‹uq ÝÑ ℓ2p rGLatinq. +Clearly }A}op “ }A:|op so that we have +››M δ ´ M 0›› +op ď }B}op ` 2 }A}op ` }D}op . +(24) +To bound }B}op we note that B is diagonal and we have +B “ +¨ +˚ +˚ +˚ +˝ +rµa +´ +1 +µδa ´ +1 +µ0a +¯ +rµb +´ +1 +µδ +b ´ +1 +µ0 +b +¯ +... +˛ +‹‹‹‚ +32 + +so that +}B}op ď +„ +max +aP r +GLatin +rµa +ˇˇˇˇ +1 +µδa +´ 1 +µ0a +ˇˇˇˇ +ȷ +“ +„ +max +aP r +GLatin +rµa +ˇˇˇˇ +1 +µδa +´ 1 +µ0a +ˇˇˇˇ +ȷ +“ +„ +max +aP r +GLatin +rµa +ˇˇˇˇ +µδ +a ´ µ0 +a +µδa ¨ µ0a +ˇˇˇˇ +ȷ +ď +„ +max +aP r +GLatin +rµa +ˇˇˇˇ +µδ +a ´ µ0 +a +rµ2a +ˇˇˇˇ +ȷ +ď +« +max +aP r +GLatin +rµa +ˇˇˇˇˇ +Kδrµp rGGreekq +rµ2a +ˇˇˇˇˇ +ff +ď δ ¨ +» +–K ¨ rµp rGGreekq +min +aP r +GLatin +µa +fi +fl . +To estimate }A}op we note +A “ +¨ +˚ +˚ +˚ +˚ +˚ +˝ +0 +⃗ηδ +apαq +µδ +a +⃗ηδ +apβq +µδ +a +¨ ¨ ¨ +0 +⃗ηδ +bpαq +µδ +b +⃗ηδ +apβq +µδ +b +¨ ¨ ¨ +0 +⃗ηδ +cpαq +µδc +⃗ηδ +cpβq +µδc +¨ ¨ ¨ +... +... +... +˛ +‹‹‹‹‹‚ +. +We can consider the map +A : ℓ2p rGGreek Y t‹uq ÝÑ ℓ2p rGLatinq. +as a composition of maps +A : ℓ2p rGGreek Y t‹uq Id +ÝÑ C| r +GGreekYt‹u| +A +ÝÑ C| r +GLatin| +Id +ÝÑ ℓ2p rGLatinq. +For the map Id : ℓ2p rGGreek Y t‹uq Ñ C| r +GGreekYt‹u| we find }Id}op “ +˜ +min +gP r +GGreekYt‹u +rµg +¸´1 +. Similarly +we find for the map Id : ℓ2p rGLatinq Ñ C| r +GLatin| that }Id}op “ +˜ +max +gP r +GLatin +rµg +¸ +. To bound the operator +norm of the map A : C| r +GGreekYt‹u| Ñ C| r +GLatin|, we use that the operator-norm is smaller than the +maximal column-sum times +b +| rGGreek Y t‹u|. Hence for A as a map from C| r +GGreekYt‹u| to C| r +GLatin| we +33 + +find +}A}op ď +b +| rGGreek Y t‹u| ¨ +¨ +˚ +˝ +1 +min +gP r +GLatin +µδg +˛ +‹‚¨ max +αP r +GGreek +» +– ÿ +aP r +GLatin +⃗ηδ +apαq +fi +fl +“ +b +| rGGreek Y t‹u| ¨ +¨ +˚ +˝ +1 +min +gP r +GLatin +µδg +˛ +‹‚¨ max +αP r +GGreek +” +⃗ζδ +‹pαq +ı +“ δ ¨ K ¨ +b +| rGGreek Y t‹u| ¨ +¨ +˚ +˝ +1 +min +gP r +GLatin +µδg ¨ min +αP r +GGreek +a +rµα +˛ +‹‚ +ď δ ¨ K ¨ +b +| rGGreek Y t‹u| ¨ +¨ +˚ +˝ +1 +min +gP r +GLatin +rµg ¨ max +αP r +GGreek +a +rµα +˛ +‹‚. +Here we estimated +max +αP r +GGreek +” +⃗ζδ +‹pαq +ı +ď +1 +min +αP r +GGreek +a +rµα +}⃗ζδ +‹}ℓ2p r +GGreekq. +In total, we find for the operator-norm of +A : ℓ2p rGGreek Y t‹uq ÝÑ ℓ2p rGLatinq. +that +}A}op ď δ ¨ K ¨ +b +| rGGreek Y t‹u| ¨ +¨ +˚ +˚ +˝ +max +gP r +GLatin +rµg +min +gP r +GLatin +rµg ¨ +max +αP r +GGreekYt‹u +rµ +3 +2α +˛ +‹‹‚. +Thus let us now investigate }D}op. +As before. +let us denote by u P ℓ2p rGGreek Y t‹uq the +restriction of an element u P ℓ2p rG to rGGreek Y t‹u. We have +}D}op “ +›››››› +ÿ +xP r +GLatinYt‹u +xψδ +x, ¨yℓ2p r +GGreekYt‹uq +µδx +ψδ +x ´ +ÿ +xP r +GLatinYt‹u +xψ0 +x, ¨yℓ2p r +GGreekYt‹uq +µ0x +ψ0 +x +›››››› +ď +›››››› +ÿ +xP r +GLatin +xψδ +x, ¨yℓ2p r +GGreekYt‹uq +µδx +ψδ +x ´ +ÿ +xP r +GLatin +xψ0 +x, ¨yℓ2p r +GGreekYt‹uq +µ0x +ψ0 +x +›››››› +` +››››› +xψδ +‹, ¨yℓ2p r +GGreekYt‹uq +µδ‹ +ψδ +‹ ´ +xψ0 +‹, ¨yℓ2p r +GGreekYt‹uq +µ0‹ +ψ0 +‹ +››››› +“ +›››››› +ÿ +xP r +GLatin +xψδ +x, ¨yℓ2p r +GGreekYt‹uq +µδx +ψδ +x +›››››› +` +››››› +xψδ +‹, ¨yℓ2p r +GGreekYt‹uq +µδ‹ +ψδ +‹ ´ +xψ0 +‹, ¨yℓ2p r +GGreekYt‹uq +µ0‹ +ψ0 +‹ +››››› . +We note for the matrix representation of the first term, that (with α, β P rGGreek Y t‹u) we have +¨ +˝ ÿ +xP r +GLatin +xψδ +x, ¨yℓ2p r +GGreekYt‹uq +µδx +ψδ +x +˛ +‚ +αβ +“ +¨ +˝ ÿ +xP r +GLatin +1 +µδx +⃗ηδ +xpαq⃗ηδ +xpβqrµβ +˛ +‚. +34 + +Using the ’maximal row sum trick’ complementary to the ’maximal column sum trick’ already used +for A above and recalling the definition of the weights +µδ +g :“ +ÿ +hP r +G +ψδ +gphq ¨ rµh +we find +›››››› +ÿ +xP r +GLatin +xψδ +x, ¨yℓ2p r +GGreekYt‹uq +µδx +ψδ +x +›››››› +ď +b +| rGGreek Y t‹u| ¨ +max +xP r +GGreekYt‹u +a +rµx +min +xP r +GGreekYt‹u +a +rµx +¨ +max +βP r +GGreekYt‹u +¨ +˝ +ÿ +αP r +GGreekYt‹u +¨ +˝ ÿ +xP r +GLatin +1 +µδx +⃗ηδ +xpαq⃗ηδ +xpβqrµβ +˛ +‚ +˛ +‚ +ď +b +| rGGreek Y t‹u| ¨ +max +xP r +GGreekYt‹u +a +rµx +min +yP r +GGreekYt‹u +a +rµy +¨ +max +αP r +GGreekYt‹u +¨ +˝ ÿ +xP r +GLatin +1 +µδx +⃗ηδ +xpαq +˛ +‚ +ď +b +| rGGreek Y t‹u| ¨ +max +xP r +GGreekYt‹u +a +rµx +min +yP r +GGreekYt‹u +a +rµy +¨ +max +αP r +GGreekYt‹u +¨ +˝ ÿ +xP r +GLatin +⃗ηδ +xpαq +˛ +‚ +ď +b +| rGGreek Y t‹u| ¨ +max +xP r +GGreekYt‹u +a +rµx +min +yP r +GGreekYt‹u +a +rµy +¨ +max +αP r +GGreekYt‹u +⃗ζδ +‹pαq +ď +b +| rGGreek Y t‹u| ¨ +max +xP r +GGreekYt‹u +a +rµx +min +yP r +GGreekYt‹u +rµ +3 +2y +¨ +max +αP r +GGreekYt‹u +}⃗ζδ +‹|ℓ2p r +GGreekq +ď +b +| rGGreek Y t‹u| ¨ +max +xP r +GGreekYt‹u +a +rµx +min +yP r +GGreekYt‹u +rµ +3 +2y +¨ K ¨ δ. +35 + +It remains to bound the second term. We find (using +›››ψδ +‹ +››› +ℓ2p r +GGreekYt‹uq ď +›››ψ0 +‹ +››› +ℓ2p r +GGreekYt‹uq): +››››› +xψδ +‹, uyℓ2p r +GGreekYt‹uq +µδ‹ +ψδ +‹ ´ +xψ0 +‹, uyℓ2p r +GGreekYt‹uq +µ0‹ +ψ0 +‹ +››››› +ℓ2p r +GGreekYt‹uq +ď +›››› +ˆ 1 +µδ‹ +´ 1 +µ0‹ +˙ +xψδ +‹, uyℓ2p r +GGreekYt‹uqψδ +‹ +›››› +ℓ2p r +GGreekYt‹uq +` 1 +µ0‹ +›››xψδ +‹, uyℓ2p r +GGreekYt‹uqψδ +‹ ´ xψ0 +‹, uyℓ2p r +GGreekYt‹uqψ0 +‹ +››› +ℓ2p r +GGreekYt‹uq +ď +ˇˇˇˇ +1 +µδ‹ +´ 1 +µ0‹ +ˇˇˇˇ ¨ +›››ψδ +‹ +››› +2 +ℓ2p r +GGreekYt‹uq ¨ }u}ℓ2p r +GGreekYt‹uq +` 1 +µ0‹ +››› +´ +xψδ +‹, uyℓ2p r +GGreekYt‹uq ´ xψ0 +‹, uyℓ2p r +GGreekYt‹uq +¯ +ψ0 +‹ ` xψδ +‹, uyℓ2p r +GGreekYt‹uq +´ +ψδ +‹ ´ ψ0 +‹ +¯››› +ℓ2p r +GGreekYt‹uq +ď +ˇˇˇˇ +1 +µδ‹ +´ 1 +µ0‹ +ˇˇˇˇ ¨ +›››ψ0 +‹ +››› +2 +ℓ2p r +GGreekYt‹uq ¨ }u}ℓ2p r +GGreekYt‹uq +`2 1 +µ0‹ +›››ψδ +‹ ´ ψ0 +‹ +››› +ℓ2p r +GGreekYt‹uq ¨ +›››ψ0 +‹ +››› +ℓ2p r +GGreekYt‹uq ¨ }u}ℓ2p r +GGreekYt‹uq +ď +¨ +˝ +δ ¨ K ¨ rµp rGGreekq +´ +rµ‹ ` rµp rGGreekq +¯ +rµ‹ +˛ +‚¨ +´ +rµ‹ ` rµp rGGreekq +¯ +¨ }u}ℓ2p r +GGreekYt‹uq +`2 +1 +rµ‹ ` rµp rGGreekq +¨ +´ +δ ¨ K ¨ rµp rGGreekq +¯ +¨ +b +rµ‹ ` rµp rGGreekq ¨ }u}ℓ2p r +GGreekYt‹uq +Thus we find +}D}op ď δ ¨ K ¨ rµp rGGreekq ¨ +¨ +˝ 1 +rµ‹ +` 2 +1 +b +rµ‹ ` rµp rGGreekq +˛ +‚ +In total, using (23) and (24), we find +›››pu ´ J rJuq +››› +ℓ2p r +Gq +ďδ +1 +2 ¨ +¨ +˝C r +GGreekYt‹u ¨ +b +rµp rGGreekq ` rµ‹ +? +Ω +˛ +‚¨ +b +E r +Gpuq +`δ ¨ +» +–K ¨ rµp rGGreekq +min +aP r +GLatin +µa +fi +fl ¨ }u}ℓ2p r +Gq ` 2 ¨ δ ¨ K ¨ +b +| rGGreek Y t‹u| ¨ +¨ +˚ +˚ +˝ +max +gP r +GLatin +rµg +min +gP r +GLatin +rµg ¨ +max +αP r +GGreekYt‹u +rµ +3 +2α +˛ +‹‹‚¨ }u}ℓ2p r +Gq +`δ ¨ K ¨ rµp rGGreekq ¨ +¨ +˝ 1 +rµ‹ +` 2 +1 +b +rµ‹ ` rµp rGGreekq +˛ +‚¨ }u}ℓ2p r +Gq +36 + +and may hence set +ϵ “ δ +1 +2 ¨ +¨ +˝C r +GGreekYt‹u ¨ +b +rµp rGGreekq ` rµ‹ +? +Ω +˛ +‚ +`δ ¨ +» +–K ¨ rµp rGGreekq +min +aP r +GLatin +µa +fi +fl ` 2 ¨ δ ¨ K ¨ +b +| rGGreek Y t‹u| ¨ +¨ +˚ +˚ +˝ +max +gP r +GLatin +rµg +min +gP r +GLatin +rµg ¨ +max +αP r +GGreekYt‹u +rµ +3 +2α +˛ +‹‹‚ +`δ ¨ K ¨ rµp rGGreekq ¨ +¨ +˝ 1 +rµ‹ +` 2 +1 +b +rµ‹ ` rµp rGGreekq +˛ +‚ +Left-hand-side of (20): +The left hand side of (20) is true with ϵ “ 0 by definition. +Right-hand-side of (20): +Let us thus check the right hand side of (20): +We have +p rJu ´ rJ1uqpxq “ 1 +µx +xψδ +x, uyℓ2p r +Gq ´ upxq. +We note +}p rJu ´ rJ1uq}ℓ2pGq ď +ˇˇˇˇ +1 +µδ‹ +xu, ψδ +‹y ´ up‹q +ˇˇˇˇ +b +µδ‹ ` +g +f +f +f +e +ÿ +xPG +g‰‹ +ˇˇˇˇ +1 +µx +xψδx, uyℓ2p r +Gq ´ upxq +ˇˇˇˇ +2 +µδx. +(25) +We first deal with the left hand term of the estimate and note that for x “ ˚ we have +µδ +‹ ď µ0 +‹ “ rµ‹ ` rµp rGGreekq +and in the limit δ Ñ 0 that +ˇˇˇˇ +1 +µδ‹ +xψδ +‹, uyℓ2p r +Gq ´ up‹q +ˇˇˇˇ ÝÑ +1 +rµ‹ ` rµp rGGreekq +ˇˇˇˇˇˇ +» +– +ÿ +gP r +GGreekYt‹u +upgq +fi +fl ´ up‹q +ˇˇˇˇˇˇ +“ +1 +rµ‹ ` rµp rGGreekq +ˇˇˇˇˇˇ +ÿ +gP r +GGreekYt‹u +upgq ´ up‹q +ˇˇˇˇˇˇ +ď +1 +rµ‹ ` rµp rGGreekq +ÿ +gP r +GGreekYt‹u +|upgq ´ up‹q| +ď +1 +rµ‹ ` rµp rGGreekq +ÿ +gP r +GGreekYt‹u +δ +1 +2 +˜ +C r +GGreekYt‹u +? +Ω +¸ b +E r +Gpuq +ďδ +1 +2 ¨ | rGGreek Y t‹u| +rµ‹ ` rµp rGGreekq +˜ +C r +GGreekYt‹u +? +Ω +¸ b +E r +Gpuq +37 + +Here we applied Lemma J.7. Comparing the δ ą 0 and δ “ 0 terms, we find +ˇˇˇˇ +1 +µδ‹ +xψδ +‹, uyℓ2p r +Gq ´ +1 +µ0‹xψ0 +‹, uyℓ2p r +Gq +ˇˇˇˇ +ď 1 +µδ‹ +ˇˇˇxψδ +‹ ´ ψ0 +‹, uyℓ2p r +Gq +ˇˇˇ ` +ˇˇˇˇ +1 +µδ‹ +´ 1 +µ0‹ +ˇˇˇˇ ¨ +ˇˇˇxψ0 +‹, uyℓ2p r +Gq +ˇˇˇ +ď 1 +rµ‹ +}u}ℓ2p r +Gq ¨ }⃗ζδ +‹}ℓ2p r +GGreekq ` +ˇˇˇˇ +1 +µδ‹ +´ 1 +µ0‹ +ˇˇˇˇ ¨ +´ +rµ‹ ` rµp rGGreekq +¯ +}u}ℓ2p r +G +ďKδ +rµ‹ +}u}ℓ2p r +Gq ` +¨ +˝ +Kδ +rµ‹ +´ +rµ‹ ` rµp rGGreekq +¯ +˛ +‚¨ +´ +rµ‹ ` rµp rGGreekq +¯ +}u}ℓ2p r +G +“δ 2K +rµ‹ +}u}ℓ2p r +Gq. +Thus we have +ˇˇˇˇ +1 +µδ‹ +xu, ψδ +‹y ´ up‹q +ˇˇˇˇ +b +µδ‹ ďδ +1 +2 ¨ +| rGGreek Y t‹u| +b +rµ‹ ` rµp rGGreekq +˜ +C r +GGreekYt‹u +? +Ω +¸ b +E r +Gpuq +`δ 2K +rµ‹ +}u}ℓ2p r +Gq. +For the remaining term in (25) we note +g +f +f +e +ÿ +xP r +GLatin +ˇˇˇˇ +1 +µx +xψδx, uyℓ2p r +Gq ´ upxq +ˇˇˇˇ +2 +ď +g +f +f +e +ÿ +xP r +GLatin +ˇˇˇˇ1 ´ rµx +µδx +ˇˇˇˇ +2 +¨ |upxq|2µδx ` +ÿ +xP r +GLatin +ˇˇˇxψδ +x, uyℓ2p r +GGreekYt‹uq +ˇˇˇ +b +µδx +ďKδ +rµ‹ +¨ }u}ℓ2p r +Gq ` +ÿ +xP r +GLatin +xψδ +x, |u|yℓ2p r +GGreekYt‹uq +b +µδx +ďKδ +rµ‹ +¨ }u}ℓ2p r +Gq ` }⃗ζδ +‹}ℓ2p r +GGreekq ¨ +„ +max +xP r +GLatin +b +µδx +ȷ +}u}ℓ2p r +Gq +ďKδ +rµ‹ +¨ }u}ℓ2p r +Gq ` δKrµp rGGreekq ¨ +„ +max +xP r +GLatin +b +Ă +µx ` δKrµp rGGreekq +ȷ +}u}ℓ2p r +Gq +ďKδ +rµ‹ +¨ }u}ℓ2p r +Gq ` δKrµp rGGreekq ¨ +«c +max +xP r +GLatin +Ă +µx ` +b +δKrµp rGGreekq +ff +}u}ℓ2p r +Gq. +Equation (21): +It finally only remains to prove the energy differences of (21) and establish +|E r +GpJ1f, uq ´ EGpf, rJ1uq| ď ϵ ¨ +a +}f}2 ` EGpfq ¨ +b +}u}2 ` E r +Gpuq. +38 + +We note that the (unique) operator associated to the energy EG via +EGpg, fq “ xg, ∆Gfyℓ2pGq +is given by +p∆Gfqpxq “ 1 +µx +ÿ +y„Gx +Wxypfpxq ´ fpyqq. +Here the notation "y „G x" signifies that nodes x and y are connected within G through edges with +positive edge-weights Wxy ą 0. +Similarly the operator associated to E r +G via +E r +Gpv, uq “ xv, ∆ r +Guyℓ2p r +Gq +is given by +p∆ r +Guqpxq “ 1 +rµx +ÿ +y„Ă +Gx +Ă +Wxypupxq ´ upyqq +with the equivalence relation „ r +G precisely signifying that Ă +Wxy ą 0. +As before. let us denote by u P ℓ2p rGGreekYt‹uq the restriction of an element u P ℓ2p rG to rGGreekYt‹u. +We note +EGpψx, uq “ xψx, ∆Guyℓ2pGq “ +ÿ +y„Gx +Wxypupxq ´ upyqq +on the smaller graph G. For the graph rG we find +E r +Gpψx, uq “ +ÿ +y„Ă +Gx +Ă +Wxypupxq ´ upyqq +` +ÿ +αP r +GGreek +⃗ηδ +xpαq +ÿ +y„Ă +Gα +Ă +Wαypupαq ´ upyqq. +Remembering that we have +J1f “ Jf “ +ÿ +xPG +fpxqψx +and p rJ1uqpxq “ upxq, +we note +ˇˇˇE r +GpJ1f, uq ´ EGpf, rJ1uq +ˇˇˇ ď +ˇˇˇˇˇˇ +ÿ +xP r +GLatinYt‹u +fpxq +“ +E r +Gpψx, uq ´ EGpψx, uq +‰ +ˇˇˇˇˇˇ +ď +¨ +˚ +˚ +˝ +1 +c +min +xP r +GLatinYt‹u +rµx +˛ +‹‹‚¨ }f}ℓ2pGq ¨ +ÿ +xP r +GLatinYt‹u +ˇˇE r +Gpψx, uq ´ EGpψx, uq +ˇˇ +Let us first bound the terms corresponding to x ‰ ‹: We have +EGpψx, uq “ +ÿ +y„Gx +y‰‹ +Wxypupxq ´ upyqq ` Wx‹pupxq ´ up‹qq +“ +ÿ +y„Gx +y‰‹ +Ă +Wxypupxq ´ upyqq ` Wx‹pupxq ´ up‹qq, +39 + +as well as +E r +Gpψx, uq “ +ÿ +y„Ă +Gx +Ă +Wxypupxq ´ upyqq ` +ÿ +αP r +GGreek +⃗ηδ +xpαq +ÿ +y„Ă +Gx +Wαypupαq ´ upyqq +“ +ÿ +y„Gx +y‰‹ +Ă +Wxypupxq ´ upyqq +`Ă +Wx‹pupxq ´ up‹qq +` +ÿ +αP r +GGreek +Ă +Wxαpupxq ´ upαqq +` +ÿ +αP r +GGreek +⃗ηδ +xpαq +ÿ +y„Ă +Gx +Ă +Wαypupαq ´ upyqq. +Hence (for x ‰ ‹) +EGpψx, uq ´ E r +Gpψx, uq “ Wx‹pupxq ´ up‹qq ´ Ă +Wx‹pupxq ´ up‹qq +´ +ÿ +αP r +GGreek +Ă +Wxαpupxq ´ upαqq +´ +ÿ +αP r +GGreek +⃗ηδ +xpαq +ÿ +y„Ă +Gx +Wαypupαq ´ upyqq +“ +¨ +˝ +ÿ +αP r +GGreek +Ă +Wxα +˛ +‚pupxq ´ up‹qq +´ +ÿ +αP r +GGreek +Ă +Wxαpupxq ´ upαqq +´ +ÿ +αP r +GGreek +⃗ηδ +xpαq +ÿ +y„Ă +Gα +Wαypupαq ´ upyqq +“ +¨ +˝ +ÿ +αP r +GGreek +Ă +Wxαpupαq ´ up‹qq +˛ +‚ +loooooooooooooooooomoooooooooooooooooon +“:Ix +´ +¨ +˝ +ÿ +αP r +GGreek +⃗ηδ +xpαq +ÿ +y„Ă +Gα +Ă +Wαypupαq ´ upyqq +˛ +‚ +loooooooooooooooooooooooooomoooooooooooooooooooooooooon +“:IIx +. +(26) +For Ix we find – using Lemma J.7 – that +|Ix| ď +¨ +˝ +ÿ +αP r +GGreek +Ă +Wxα +˛ +‚¨ δ +1 +2 +˜ +C r +GGreekYt‹u +? +Ω +¸ b +E r +Gpuq +and hence +ÿ +xPG +x‰‹ +|Ix| ď +¨ +˚ +˝ +ÿ +xPG +x‰‹ +ÿ +αP r +GGreek +Ă +Wxα +˛ +‹‚¨ δ +1 +2 +˜ +C r +GGreekYt‹u +? +Ω +¸ b +E r +Gpuq. +40 + +To bound |IIx| we note +ˇˇˇˇˇˇ +ÿ +αP r +GGreek +⃗ηδ +xpαq +ÿ +y„Ă +Gα +Ă +Wαypupαq ´ upyqq +ˇˇˇˇˇˇ +“ +ˇˇˇˇˇˇ +ÿ +αP r +GGreek +⃗ηδ +xpαq +ÿ +y„Ă +Gα +b +Ă +Wαy +b +Ă +Wαypupαq ´ upyqq +ˇˇˇˇˇˇ +“ +ˇˇˇˇˇˇˇ +ÿ +αP r +GGreek +⃗ηδ +xpαq +» +– ÿ +y„Ă +Gα +Ă +Wyα +fi +fl +1 +2 +¨ +» +– ÿ +y„Ă +Gα +Ă +Wyα|upαq ´ upyq|2 +fi +fl +1 +2 ˇˇˇˇˇˇˇ +ď +ÿ +αP r +GGreek +⃗ηδ +xpαq ¨ +» +– ÿ +y„Ă +Gα +Ă +Wyα +fi +fl +1 +2 +¨ +b +E r +Gpuq. +Thus we find – using Cauchy-Schwarz – that +ÿ +xPG +x‰‹ +|IIx| ď +ÿ +xPG +x‰‹ +ÿ +αP r +GGreek +⃗ηδ +xpαq ¨ +» +– ÿ +y„Ă +Gα +Ă +Wyα +fi +fl +1 +2 +¨ +b +E r +Gpuq +“ +ÿ +αP r +GGreek +⃗ζδ +‹pαq ¨ +» +– ÿ +y„Ă +Gα +Ă +Wyα +fi +fl +1 +2 +¨ +b +E r +Gpuq +ď +1 +min +αP r +GGreek +a +rµα +¨ }⃗ζδ +‹}ℓ2p r +GGreekq ¨ +» +– +ÿ +αP r +GGreek +ÿ +y„Ă +Gα +Ă +Wyα +fi +fl +1 +2 +¨ +b +E r +Gpuq +ď +1 +min +αP r +GGreek +a +rµα +¨ Kδ ¨ +» +– +ÿ +αP r +GGreek +ÿ +y„Ă +Gα +Ă +Wyα +fi +fl +1 +2 +¨ +b +E r +Gpuq +ď +1 +min +αP r +GGreek +a +rµα +¨ Kδ ¨ +d +ÿ +αP r +GGreek +rdα ¨ +b +E r +Gpuq. +Here we denoted by rdα the degree of the node α. We further note +ÿ +αP r +GGreek +rdα “ +ÿ +αP r +GGreek +ÿ +yP r +GLatin +Ă +Wαy ` 1 +δ +ÿ +αP r +GGreek +ÿ +yP r +GGreekYt‹u +ωαy. +Writing +rd1 +int :“ +ÿ +αP r +GGreek +ÿ +yP r +GGreekYt‹u +ωαy +for the sum of ’internal’ degrees of greek nodes within Greek Y t‹u at δ “ 1 and +dexternal :“ +ÿ +αP r +GGreek +ÿ +yP r +GLatin +Ă +Wαy +for the ’total connection strength’ between the Greek and Latin sector, we thus find +ÿ +xPG +x‰‹ +|IIx| ď r +b +rd1 +int ¨ +? +δ ` +a +dexternal ¨ δs +K +min +αP r +GGreek +a +rµα +¨ +b +E r +Gpuq. +It remains to bound the x “ ‹ term in (26). To this end we note +EGpψ‹, uq “ +ÿ +y„G‹ +W‹ypup‹q ´ upyqq +41 + +and +E r +Gpψ‹, uq “ +ÿ +y„Ă +G‹ +Ă +W‹ypup‹q ´ upyqq +` +ÿ +αP r +GGreek +⃗ζδ +‹pαq +ÿ +y„Ă +Gα +Ă +Wyαpupαq ´ upyqq. +For the difference of the energy forms we thus find +EGpψ‹, uq ´ E r +Gpψ‹, uq “ +ÿ +y„G‹ +W‹ypup‹q ´ upyqq +´ +ÿ +y„Ă +G‹ +Ă +W‹ypup‹q ´ upyqq ´ +ÿ +αP r +GGreek +⃗ηδ +‹pαq +ÿ +y„Ă +Gα +Ă +Wyαpupαq ´ upyqq +“ +ÿ +y„G‹ +W‹ypup‹q ´ upyqq +´ +ÿ +y„Ă +G‹ +Ă +W‹ypup‹q ´ upyqq ´ +ÿ +αP r +GGreek +⃗ηδ +‹pαq +ÿ +y„Ă +Gα +Ă +Wyαpupαq ´ upyqq +` +ÿ +αP r +GGreek +ÿ +y„Ă +Gα +Ă +Wyαpupαq ´ upyqq ´ +ÿ +αP r +GGreek +ÿ +y„Ă +Gα +Ă +Wyαpupαq ´ upyqq. +We have +ÿ +αP r +GGreek +ÿ +y„Ă +Gα +Ă +Wyαpupαq ´ upyqq “ +ÿ +αP r +GGreek +Ă +W‹αpupαq ´ up‹qq ` +ÿ +y„Ă +Gα +yP r +GLatin +Ă +Wyαpupαq ´ upyqq +` +ÿ +αP r +GGreek +ÿ +y„Ă +Gα +yP r +GGreek +Ă +Wyαpupαq ´ upyqq +loooooooooooooooooooomoooooooooooooooooooon +“0 +. +42 + +with the last term vanishing by symmetry. This implies +EGpψ‹, uq ´ E r +Gpψ‹, uq “ +ÿ +αP r +GGreek +p1 ´ ⃗ηδ +‹pαqq +ÿ +y„Ă +Gα +Ă +Wyαpupαq ´ upyqq +` +ÿ +y„G‹ +yP r +GLatin +¨ +˝ +ÿ +αP r +GGreekYt‹u +Ă +Wyα +˛ +‚pup‹q ´ upyqq +´ +ÿ +y„Ă +G‹ +Ă +W‹ypup‹q ´ upyqq +´ +ÿ +αP r +GGreek +Ă +W‹αpupαq ´ up‹qq ´ +ÿ +αP r +GGreek +ÿ +y„Ă +Gα +yP r +GLatin +Ă +Wαypupαq ´ upyqq +“ +ÿ +αP r +GGreek +p1 ´ ⃗ηδ +‹pαqq +ÿ +y„Ă +Gα +Ă +Wyαpupαq ´ upyqq +` +ÿ +y„Ă +G‹ +yP r +GLatin +¨ +˝ +ÿ +αP r +GGreekYt‹u +Ă +Wyα +˛ +‚pup‹q ´ upyqq +´ +ÿ +y„G‹ +yP r +GGreek‹ +Ă +W‹ypup‹q ´ upyqq +´ +ÿ +αP r +GGreek +Ă +W‹αpupαq ´ up‹qq ´ +ÿ +αP r +GGreek +ÿ +y„Ă +Gα +yP r +GLatin +Ă +Wαypupαq ´ upyqq +“ +ÿ +αP r +GGreek +p1 ´ ⃗ηδ +‹pαqq +ÿ +y„Ă +Gα +Ă +Wyαpupαq ´ upyqq +` +ÿ +y„G‹ +yP r +GLatin +¨ +˝ +ÿ +αP r +GGreekYt‹u +Ă +Wyα +˛ +‚pup‹q ´ upyqq +´ +ÿ +αP r +GGreek +ÿ +y„Ă +Gα +yP r +GLatin +Ă +Wαypupαq ´ upyqq. +Continuing, we find +EGpψ‹, uq ´ E r +Gpψ‹, uq “ +ÿ +αP r +GGreek +p1 ´ ⃗ηδ +‹pαqq +ÿ +y„Ă +Gα +Ă +Wyαpupαq ´ upyqq +` +ÿ +αP r +GGreek +ÿ +y„G‹ +yP r +GLatin +Ă +Wyαpup‹q ´ upyqq +´ +ÿ +αP r +GGreek +ÿ +y„Ă +Gα +yP r +GLatin +Ă +Wαypupαq ´ upyqq +“ +ÿ +αP r +GGreek +p1 ´ ⃗ηδ +‹pαqq +ÿ +y„Ă +Gα +Ă +Wyαpupαq ´ upyqq +` +ÿ +αP r +GGreek +ÿ +y„Ă +Gα +yP r +GLatin +Ă +Wyαpup‹q ´ upyqq +´ +ÿ +αP r +GGreek +ÿ +y„Ă +Gα +yP r +GLatin +Ă +Wyαpupαq ´ upyqq. +43 + +This – in turn – we can write as +EGpψ‹, uq ´ E r +Gpψ‹, uq “I ` II +with +I :“ +ÿ +αP r +GGreek +⃗ζδ +‹pαq +ÿ +y„Ă +Gα +Ă +Wyαpupαq ´ upyqq, +and +II :“ +ÿ +αP r +GGreek +ÿ +y„Ă +Gα +yP r +GLatin +Ă +Wyαpup‹q ´ upαqq. +For the first term, we find +|I| ď +}⃗ζδ +‹} +min +αP r +GGreek +a +rµα +¨ +d +ÿ +αP r +GGreek +rdα ¨ +b +E r +Gpuq +ďr +b +rd1 +int ¨ +? +δ ` +a +dexternal ¨ δs +K +min +αP r +GGreek +a +rµα +¨ +b +E r +Gpuq. +For the second term we note +|II| ď +ÿ +αP r +GGreek +ÿ +y„Ă +Gα +yP r +GLatin +Ă +Wyα|up‹q ´ upαq| +ď +? +δ ¨ +ÿ +αP r +GGreek +ÿ +y„Ă +Gα +yP r +GLatin +Ă +Wyα +˜ +C r +GGreekYt‹u +? +Ω +¸ b +E r +Gpuq +“ +? +δ ¨ dexternal ¨ +˜ +C r +GGreekYt‹u +? +Ω +¸ b +E r +Gpuq. +K +PROOF OF THEOREM 5.7 +We prove the following theorem: +Theorem K.1. In the setting of Theorem 5.6 denote by T ( rT) adjacency matrices or normalized +graph Laplacians on ℓ2pGq (ℓ2pGq). There are no functions η1, η2 : r0, 1s Ñ Rě0 with ηipδq Ñ 0 +as δ Ñ 0 (i “ 1, 2), families of identification operators Jδ, rJδ and ω P C so that Jδ and rJδ are +η1pδq-quasi-unitary with respect to rT, T and ω while the operators rT and T remain ω-η2pδq close. +Proof. We prove these two result through contradiction on a graph with two vertices and one edge +with weight 1{δ, which we collapse. +First fix T ( rT) to be the adjacency matrices +Ă +W “ +ˆ +0 +1 +δ +1 +δ +0 +˙ +and +W “ 0. +The eigenvectors and eigenvalues of Ă +W are given by t´ 1 +δ , 1 +δ u and +v´ “ +ˆ +1 +´1 +˙ +and v` “ +ˆ +1 +1 +˙ +. +44 + +Denote the orthogonal projections onto the corresponding eigenspaces by tP´, P`u. Take the +function g to be defined as +gpλq :“ 1 ´ +i +i ´ λ. +Then since gp0q “ 0 we have +gpWq “ 0. +Furthermore we have +gpĂ +Wq “ +„ +1 ´ +i +i ´ 1 +δ +ȷ +P` ` +„ +1 ´ +i +i ` 1 +δ +ȷ +P´ +“ P` ` P´ ´ δ +1 +δ ` iP` ´ δ +1 +δ ´ iP´ +“ Id ´ δ +1 +δ ` iP` ´ δ +1 +δ ´ iP´ +“ Id +„ +1 ´ δ +1 +δ ` i +ȷ +` +„ +δ +1 +δ ` i ´ δ +1 +δ ´ i +ȷ +P´ +“ Id +„ +1 ´ δ +1 +δ ` i +ȷ +´ +„ +δ +2i +δ2 ` 1 +ȷ +P´ +We are interested in +›››gpĂ +WqJδ ´ JδgpWq +››› +op “ +›››gpĂ +WqJδ››› +op “ +››››Jδ ´ δ +„ +1 +δ ` iP` ` +1 +δ ´ iP´ +ȷ +Jδ +›››› +op +. +Assuming +›››gpĂ +WqJδ ´ JδgpWq +››› +op “ +›››gpĂ +WqJδ››› +op ď η1pδq +we also find +ˇˇˇˇ +››Jδ›› +op +ˆ +i +δ ` i +˙ +´ +››JδP´ +›› +op +ˆ δ2i +δ2 ` 1 +˙ˇˇˇˇ ď η1pδq. +Thus also +››Jδ›› +op +ˆ +i +δ ` i +˙ +ď η1pδq ` +››JδP´ +›› +op +ˆ δ2i +δ2 ` 1 +˙ +. +Taking the limit and using the condition }Jδ}op ď 2, we find that +››Jδ›› Ñ 0 as δ Ñ 0. Since we +demand +}pJ ´ rJ˚q}op ď η2pδq +with +lim +δÑ0 η2pδq “ 0, +we also find } rJ}op “ } rJ˚}op Ñ 0. Next we note that we have +Rω “ 1 +ω +and demand +}pId ´ rJδJδqRω}op Ñ 0. +However +}pId ´ rJδJδqRω}op “ 1 +|ω|}Id ´ rJδJδ}op ě 1 +|ω|p1 ´ } rJ˚}op}J}opq Ñ 1 +|ω| ą 0. +45 + +Thus we have our contradiction. +Hence let us now choose T ( rT) as the normalized graph Laplacians associated to the adja- +cency matrices W (Ă +W) from above. We thus have +L “ 0 +and +Ă +L “ +ˆ +1 +´1 +´1 +1 +˙ +. +The eigenvectors and eigenvalues of Ă +L are given by t0, 2u and +v0 “ +ˆ +1 +1 +˙ +and v2 “ +ˆ +1 +´1 +˙ +. +Denote the orthogonal projections onto the corresponding eigenspaces by tP0, P2u. Then +Ă +L “ 2P2. +Chose a function g such that gp0q “ 0 and without loss of generality assume gp2q “ 1. Then +0 ÐÝ +›››gp Ă +L qJδ ´ JδgpL q +››› +op “ +››P2Jδ›› +op . +(27) +Next we consider the demand +}pId ´ Jδ rJδq rRωu} ď η3 ¨ }u}. +(28) +Since p Ă +L ´ ωIdq is bijective, (28) is implies +}pId ´ Jδ rJδqv} ď η3pδq ¨ r|ω|}v} ` } Ă +L } ¨ }v}s “ η3pδq ¨ r|ω| ` 2s ¨ }v}. +(29) +upon writing +u “ p Ă +L ´ ωIdqv. +We also write +v “ +ˆ +va +vb +˙ +. +We write +rJδ “ +ˆ +aδ +bδ +˙T +and +Jδ “ η4pδq ¨ +ˆ +1 +´1 +˙ +` fpδq ¨ +ˆ +1 +1 +˙ +. +From (27), we know that +lim +δÑ0 η4pδq “ 0, +but we do not yet know the behaviour of fp¨q, aδ, bδ as δ Ñ 0. +With the above notation, we find from (29) that +}pId ´ Jδ rJδqv} “ +›››› +ˆ +va ´ fpδqaδva ´ fpδqbδvb +vb ´ fpδqaδva ´ fpδqbδvb +˙ +´ η4pδq +Bˆ +va +vb +˙ +, +ˆ +aδ +bδ +˙F ˆ +1 +1 +˙›››› +ě +›››› +ˆ +va ´ fpδqaδva ´ fpδqbδvb +vb ´ fpδqaδva ´ fpδqbδvb +˙›››› ´ η4pδq ¨ 4 ¨ }v}. +46 + +Thus, combining this result with (29), we know that +›››› +ˆ +va ´ fpδqaδva ´ fpδqbδvb +vb ´ fpδqaδva ´ fpδqbδvb +˙›››› ÝÑ 0. +Thus, since both entries of the above vector need to tend to zero, we need both +fpδq ¨ aδ Ñ 1 and fpδq ¨ bδ Ñ 0 +as well as +fpδq ¨ aδ Ñ 0 and fpδq ¨ bδ Ñ 1 +which yields the desired contradiction. +L +PROOF OF THEOREM 5.8 +We first note how the graph Laplacian ∆GN as we have defined it, is consistent with the underlying +positive (in the sense of non-negative eigenvalues) Laplacian +” ´ ∆S1 “ ´ B2 +Bθ2 ” +on the unit circle S1. +To this end, fix 0 ă h ăă 1. Fix a point x P S1. For any suitable function f – by means of Taylor +expansions – we may write +fpx ` hq “ fpxq ` h ¨ rBθfspxq ` h2 +2 ¨ r∆S1fspxq ` Oph3q +fpx ´ hq “ fpxq ´ h ¨ rBθfspxq ` h2 +2 ¨ r∆S1fspxq ` Oph3q. +Adding these two terms, we find +r´∆S1fspxq “ 2fpxq ´ fpx ` hq ´ fpx ´ hq +h2 +` Ophq. +This motivates setting our edgeweights on GN to 1{h2 with h “ 2π{N the distance between evenly +spaced nodes on the unit-circle S1. +Remark L.1. It should be noted that this consistency property – while given a heuristic to choose +weights – does not (immediately) imply ’convergence’ of ∆GN to ´∆S1 in the sense needed to e.g. +apply Levie et al. (2019a). As our proof of Theorem L proceeds completely without reference to the +limit-circle, we do not proceed beyond the above heuristic in investigating in what (relevant) sense +∆GN approximates ´∆S1. +We thus now want to prove the following result: +Theorem L.2. In the large graph setting of Section 5.2 choose all node-weights equal to one and +N to be odd for definiteness. There exists constants K1, K2 “ Op1q so that for each N ě 1, there +exist identification operators J, rJ mapping between ℓ2pGNq and ℓ2pGN`1q so that J and rJ are +pK1{Nq-quasi-unitary with respect to ∆GN , ∆GN`1 and ω “ p´1q. Furthermore, the operators +∆GN and ∆GN`1 are p´1q-pK2{Nq close with identification operator J. +Proof. We first note that the normalized eigenvectors of GN are given by +φN +k pxq “ +1 +? +N +ei 2πk +N x 0 ď k ă N. +The corresponding eigenvalues are easily found to be +λN +k “ N 2 +π2 sin2 ´ π +N ¨ k +¯ +. +47 + +For definiteness, we have assumed N to be odd, so that pN ` 1q is even. We define the identification +operator J : ℓ2pGNq Ñ ℓ2pGN`1q via +JpφN +k pxqq “ +"φN`1 +k +for K ă N +2 +φN`1 +k`1 +for K ă N +2 +on the orthonormal basis tφN +k u0ďkăN and extend it to all of ℓ2pGNq via normality. This implies that +precisely the eigenspace spanned by φN`1 +N`1 +2 +(corresponding to the eigenvalue λN`1 +N`1 +2 +“ pN ` 1q2{π2 ) +does not lie in the image of J. We set rJ to be the adjoint J˚ of J. Choosing ω “ 1, we shall now +first check the equations of Definition 5.1. Since J is isometric, we have +}Jf} “ }f} ď 2}f} +as desired. Since rJ “ J˚, we have +} rJ ´ J˚} “ 0. +Since rJJ “ Idℓ2pGNq, what remains to be checked is the demand +}pId ´ J rJq rR´1}op ď K ¨ 1 +N 2 . +We have +}pId ´ J rJq rR´1}op “ 1 ¨ +1 +1 ` λN`1 +N`1 +2 +“ +1 +1 ` N 2{π2 ď +π2 +pN ` 1q2 ď π2 ¨ 1 +N 2 . +Thus let us now check that the conditions of Definition 5.2 are fulfilled. +We note that with +our identification operator and by symmetry (λN +k “ λN +N´k), we have +}JR´1 ´ rR´1J}op “ max +0ďkă N +2 +ˇˇˇˇˇˇ +1 +1 ` N 2 +π2 sin2 ` π +N k +˘ ´ +1 +1 ` pN`1q2 +π2 +sin2 ´ +π +pN`1qk +¯ +ˇˇˇˇˇˇ +. +We now need to bound the right hand side uniformly in k as N Ñ 8. To this end we write a :“ 1{N +(which implies N`1 +N +“ 1 ` a) and x “ k +N (which for our allowed values of k implies 0 ď x ă 1 +2). +With this we have +48 + +ˇˇˇˇˇˇ +1 +1 ` N 2 +π2 sin2 ` π +N k +˘ ´ +1 +1 ` pN`1q2 +π2 +sin2 ´ +π +pN`1qk +¯ +ˇˇˇˇˇˇ +“ pπaq2 +ˇˇˇˇˇˇ +1 +pπaq2 ` sin2 pπxq ´ +1 +pπaq2 ` p1 ` aq2 sin2 ´ +πx +1 +1`a +¯ +ˇˇˇˇˇˇ +“ pπaq2 +ˇˇˇˇˇˇ +p1 ` aq2 sin2 ´ +πx +1 +1`a +¯ +´ sin2 pπxq +rpπaq2 ` sin2 pπxqs ¨ rpπaq2 ` p1 ` aq2 sin2 ´ +πx +1 +1`a +¯ +s +ˇˇˇˇˇˇ +“ pπaq2 +ˇˇˇˇˇˇ +sin2 ´ +πx +1 +1`a +¯ +´ sin2 pπxq ` a sin2 ´ +πx +1 +1`a +¯ +` a2 sin2 ´ +πx +1 +1`a +¯ +rpπaq2 ` sin2 pπxqs ¨ rpπaq2 ` p1 ` aq2 sin2 ´ +πx +1 +1`a +¯ +s +ˇˇˇˇˇˇ +“ pπaq2 +ˇˇˇˇˇˇ +sin +´ +πx +a +1`a +¯ +¨ sin +´ +πx a`2 +a`1 +¯ +` a sin2 ´ +πx +1 +1`a +¯ +` a2 sin2 ´ +πx +1 +1`a +¯ +rpπaq2 ` sin2 pπxqs ¨ rpπaq2 ` p1 ` aq2 sin2 ´ +πx +1 +1`a +¯ +s +ˇˇˇˇˇˇ +ď pπaq2 +ˇˇˇˇˇˇ +sin +´ +πx +a +1`a +¯ +¨ sin +´ +πx a`2 +a`1 +¯ +` a sin2 ´ +πx +1 +1`a +¯ +` a2 sin2 ´ +πx +1 +1`a +¯ +rsin2 ´ +πx +a +1`a +¯ +s ¨ rpπaq2s +ˇˇˇˇˇˇ +ď a +ˇˇˇˇˇˇˇ +sinpπx +a +1`aq +a +¨ sin +´ +πx a`2 +a`1 +¯ +` sin2 ´ +πx +1 +1`a +¯ +` a sin2 ´ +πx +1 +1`a +¯ +sin2 ´ +πx +1 +1`a +¯ +ˇˇˇˇˇˇˇ +ď2a ` a +ˇˇˇˇˇˇ +sin +´ +πx a`2 +a`1 +¯ +sin +´ +πx +1 +1`a +¯ +ˇˇˇˇˇˇ +¨ +ˇˇˇˇˇˇ +sin +´ +πx +a +1`a +¯ +a ¨ sin +´ +πx +1 +1`a +¯ +ˇˇˇˇˇˇ +. +Thus we are done if we can show that the function +Fpa, xq “ +ˇˇˇˇˇˇ +sin +´ +πx a`2 +a`1 +¯ +sin +´ +πx +1 +1`a +¯ +ˇˇˇˇˇˇ +¨ +ˇˇˇˇˇˇ +sin +´ +πx +a +1`a +¯ +a ¨ sin +´ +πx +1 +1`a +¯ +ˇˇˇˇˇˇ +is bounded on the rectangle r0, 1s ˆ r0, 1 +2s. We change variables y “ πx{p1 ` aq and consider +Fpa, yq “ +ˇˇˇˇ +sin pypa ` 2qq +sin pyq +ˇˇˇˇ ¨ +ˇˇˇˇ +sin pyaq +a ¨ sin pyq +ˇˇˇˇ +on r0, 1s ˆ r0, π +2 s instead. Away from y “ 0 this is obvious. Close to y “ 0 we might Taylor expand +in numerators and denominators respectively and then (formally) divide them both respectively by y +to see that the function Fpa, yq is indeed regular at y “ 0 too and hence on the entire compact set +r0, 1s ˆ r0, π +2 s. As a continuous function, F attains its supremum on this set. Denote it by K. Hence +we now know +}JR´1 ´ rR´1J}op ď r2 ` Ks ¨ a ” r2 ` Ks ¨ 1 +N . +Thus we have established the desired Op1{Nq-decay. +M +PROOF OF THEOREM 6.1 +Theorem M.1. For p ě 2 we have in the setting of Theorem 3.1 that }Ψp +Npfq ´ Ψp +Nphq}RKout ď +´śN +n“1 LnRnBn +¯ +¨ }f ´ h}Lin. In the setting of Theorem 4.3 or 5.4 and under the additional +assumption that the ’final’ identification operator JN satisfies +ˇˇ}JNfi}ℓkp r +GNq ´ }fi}ℓkpGNq +ˇˇ ď +49 + +δ ¨ K ¨ }fi}ℓ2pGNq for all fi P ℓ2pGNq, we have }Ψp +Npfq ´ rΨp +NpJ0fq}RKout ď pN ¨ DRL ` K ¨ +pBRLqq ¨ pBRLqN´1 ¨ }f}Lin ¨ δ. +Proof. To prove the first claim, we note +}Ψp +Npfq ´ Ψp +Npgq}RKout “ +d ÿ +iPKout +ˇˇ}rΦNpfqsi}ℓppGoutq ´ }rΦNpgqsi}ℓppGoutq +ˇˇ2 +ď +d ÿ +iPKout +ˇˇ}rΦNpfqsi ´ rΦNpgqsi}ℓppGoutq +ˇˇ2 +ď +d ÿ +iPKout +ˇˇ}rΦNpfqsi ´ rΦNpgqsi}ℓ2pGoutq +ˇˇ2 +“ }Φp +Npfq ´ Φp +Npgq}RKout +where we used the reverse triangle inequality and the fact that } ¨ }ℓpp r +Goutq ď } ¨ }ℓ2p r +Goutq for 2 ď p. To +finish the proof we now only need to apply Theorem 3.1. +To prove the second claim we note +}Ψp +Npfq ´ rΨp +NpJ0fq}RKout +“ +d ÿ +iPKout +ˇˇˇ}rΦNpfqsi}ℓppGoutq ´ }rrΦNpJ0fqsi}ℓpp r +Goutq +ˇˇˇ +2 +“ +d ÿ +iPKout +ˇˇˇ}rΦNpfqsi}ℓppGoutq ´ }rJNΦNpfqsi}ℓppGoutq ` }rJNΦNpfqsi}ℓppGoutq ´ }rrΦNpJ0fqsi}ℓpp r +Goutq +ˇˇˇ +2 +ď +d ÿ +iPKout +ˇˇˇ}rΦNpfqsi}ℓppGoutq ´ }JNrrΦNpfqsi}ℓppGoutq +ˇˇˇ +2 +` +d ÿ +iPKout +ˇˇˇ}JNrΦNpfqsi}ℓppGoutq ´ }rrΦNpJ0fqsi}ℓpp r +Goutq +ˇˇˇ +2 +ď K ¨ δ ¨ }JNΦpfq} Ă +Lout ` }rΦpJ0fq ´ JNΦpfq} Ă +Lout +and the claim follows as before. +The proof of the third claim proceed in complete analogy. +N +ADDITIONAL DETAILS ON EXPERIMENTAL SETUP +Scaling Operators: +The adjacency matrix fo the given graph is given by +A “ +¨ +˚ +˚ +˚ +˝ +0 +16 +7 +18 +19 +16 +0 +6 +22 +3 +7 +6 +0 +1 +90 +18 +22 +1 +0 +23 +19 +3 +90 +23 +0 +˛ +‹‹‹‚. +(30) +Collapsing Edges: +We consider the setting introduced in Section 5.2 and consider a generic fully +connected graph rG with | rG| “ 8. We consider a splitting into rG “ rGLatin +Ť rGGreek +Ťt‹u with +| rGLatin| “ 3 and | rGGreek| “ 4 . As described in Section 5.2, we assume Ą +Wab, Ă +Wa‹ “ Op1q, @a, b P +rGLatin and Ă +Wαβ “ ωαβ +δ +and Ă +Wα‹ “ ωα‹ +δ +such that pωαβ, ωα‹ “ Op1q for all α, β P rGGreek. For +completeness and reproducibility, the full adjacency matrix Ă +W can be found in Appendix N. We set +50 + +node weight on rG to one and – as discussed – construct a graph G with |G| “ 4 through ’collapsing +strong edges’. +The adjacency matrix of the larger ’un-collapsed’ graph rG we consider in Section 7 is given as +follows +Ă +W “ +¨ +˚ +˚ +˚ +˚ +˚ +˚ +˚ +˚ +˝ +0 +4 +2 +10 +4 +5 +6 +7 +4 +0 +17 +9 +8 +9 +10 +11 +2 +17 +0 +42 +12 +13 +14 +15 +10 +9 +42 +0 +16{δ +7{δ +18{δ +19{δ +4 +8 +12 +16{δ +0 +6{δ +22{δ +3{δ +5 +9 +13 +7{δ +6{δ +0 +1{δ +90{δ +6 +10 +14 +18{δ +22{δ +1{δ +0 +23{δ +7 +11 +15 +19{δ +3{δ +90{δ +23{δ +0 +˛ +‹‹‹‹‹‹‹‹‚ +(31) +The exceptional vertex ‹ here carries index "4" ("‹ “ 4"). Node weights are set to unity. +The Realm of Large Graphs: +We also plot the difference in characteristic operators as opposed to +their resolvents: +Figure 10: Operator Differences +Their distances does not decay. +Experiments on Molecules: +The dataset we consider is the QM7 dataset, introduced in Blum & +Reymond (2009); Rupp et al. (2012). This dataset contains descriptions of 7165 organic molecules, +each with up to seven heavy atoms, with all non-hydrogen atoms being considered heavy. A molecule +is represented by its Coulomb matrix CClmb, whose off-diagonal elements +CClmb +ij +“ +ZiZj +|Ri ´ Rj| +correspond to the Coulomb-repulsion between atoms i and j, while diagonal elements encode a +polynomial fit of atomic energies to nuclear charge Rupp et al. (2012): +CClmb +ii +“ 1 +2Z2.4 +i +For each atom in any given molecular graph, the individual Cartesian coordinates Ri and the atomic +charge Zi are also accessible individually. To each molecule an atomization energy - calculated via +density functional theory - is associated. The objective is to predict this quantity, the performance +metric is mean absolute error. Numerically, atomization energies are negative numbers in the range +´600 to ´2200. The associated unit is rkcal/mols. +51 + +106 +105 +Operator Differences +104 +II△Gn+1J- J△Gn llop +103 +102 +101 +100 +0 +250 +500 +750 +1000 +1250 +1500 +1750 +2000 +NO +NOTATIONAL CONVENTIONS +We provide a summary of employed notational conventions: +Table 1: Classification Accuracies on Social Network Datasets +Symbol +Meaning +G +a graph or a vertex set +|G| +number of nodes in G +µi +weight of node i +M +weight matrix +x¨, ¨y +inner product +W +adjacency matrix +D +degree matrix +∆ +graph Laplacian +L +normalized graph Laplacian +T +generic operator +T ˚ +adjoint of T +σpTq +spectrum (i.e. collection of eigenvalues) of T +λ +an eigenvalue +gpTq +function g applied to operator T +} ¨ }op +operator norm (i.e. spectral norm) +} ¨ }F +Frobenius norm +ω +a complex number +ω +complex conjugate of ω +z +a complex number +Bϵpωq +open ball of radius ϵ around ω +ag +k,bg +k +complex number determined by g and indexed by k +U +open set extending to infinity in C +D +a Cauchy domain in C +BD +the boundary of D +pωId´Tq´1, Rω +the resolvent of T at ω +γT p¨q +resolvent profile of T +ű +...dz +a complex line integral +ű +...d|z| +the corresponding real line integral +ρ +a non-linearity +P +a connecting operator +L +(possibly hidden) feature space associated to a GCN +Φ +map associated to a GCN +ϵ, δ +small numbers +J +an identification operator (possibly dependent on some ϵ +or δ) +rG +Graph consisting of regular nodes, an exceptional node +and a strongly connected sub-graph +rGGreek +nodes in a strongly connected sub-graph +‹ +exceptional node to which a strongly connected sub-graph +is collapsed +rGLatin +regular nodes in rG +EGp¨q +Energy form associated to the (undirected) graph G +h +distance between nodes on the circle +} ¨ }p +the p-norm on Rd +p +a natural number +Ψ +graph-level feature map associated to a GCN +Zi +atomic charge of atom corresponding to node i +xi +Cartesian position of atom corresponding to node i +ZiZj +}xi´xj} +Coulomb interaction between atoms i and j +}xi ´ xj} +Euclidean distance between xi and xj +52 + diff --git a/INFJT4oBgHgl3EQfFiwP/content/tmp_files/load_file.txt b/INFJT4oBgHgl3EQfFiwP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d94079106ab019ddf1f1792e4f0e86c11d967091 --- /dev/null +++ b/INFJT4oBgHgl3EQfFiwP/content/tmp_files/load_file.txt @@ -0,0 +1,2444 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf,len=2443 +page_content='LIMITLESS STABILITY FOR GRAPH CONVOLUTIONAL NETWORKS Christian Koke Technical University of Munich christian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='koke@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='de ABSTRACT This work establishes rigorous, novel and widely applicable stability guarantees and transferability bounds for graph convolutional networks – without reference to any underlying limit object or statistical distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Crucially, utilized graph- shift operators (GSOs) are not necessarily assumed to be normal, allowing for the treatment of networks on both directed- and for the first time also undirected graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Stability to node-level perturbations is related to an ’adequate (spectral) covering’ property of the filters in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Stability to edge-level perturbations is related to Lipschitz constants and newly introduced semi-norms of filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Results on stability to topological perturbations are obtained through recently developed mathematical- physics based tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' As an important and novel example, it is showcased that graph convolutional networks are stable under graph-coarse-graining procedures (replacing strongly-connected sub-graphs by single nodes) precisely if the GSO is the graph Laplacian and filters are regular at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' These new theoretical results are supported by corresponding numerical investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 1 INTRODUCTION Graph Convolutional Networks (GCNs) (Kipf & Welling, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Hammond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Defferrard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=', 2016) generalize Euclidean convolutional networks to the graph setting by replacing con- volutional filters by functional calculus filters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' scalar functions applied to a suitably chosen graph-shift-oprator capturing the geometry of the underlying graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' A key concept in trying to understand the underlying reasons for the superior numerical performance of such networks on graph learning tasks (as well as a guiding principle for the design of new architectures) is the concept of stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' In the Euclidean setting, investigating stability essentially amounts to exploring the variation of the output of a network under non-trivial changes of its input (Mallat, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Wiatowski & Bölcskei, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' In the graph-setting, additional complications are introduced: Not only input signals, but now also the graph shift operators facilitating the convolutions on the graphs may vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Even worse, there might also occur changes in the topology or vertex sets of the investigated graphs – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' when two dissimilar graphs describe the same underlying phenomenon – under which graph convolutional networks should also remain stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' This last stability property is often also referred to as transferability (Levie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=', 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Previous works investigated stability under changes in graph-shift operators for specific filters (Levie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=', 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Gama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=', 2020) or the effect of graph-rewiring when choosing a specific graph shift operator (Kenlay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Stability to topological perturbations has been established for (large) graphs discretising the same underlying topological space (Levie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=', 2019a), the same graphon (Ruiz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Maskey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=', 2021) or for graphs drawn from the same statistical distribution (Keriven et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Common among all these previous works are two themes limiting practical applicability: First and foremost, the class of filters to which results are applicable is often severely restricted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' The same is true for the class of considered graph shift operators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' with non-normal operators (describing directed graphs) either explicitly or implicitly excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Furthermore – when investigating transferability properties – results are almost exclusively available under the assumption that graphs are large and either discretize the same underlying ’continuous’ limit object suffieciently well, or are drawn from the same statistical distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' While these are of course relevant regimes, they do not allow to draw conclusions beyond such asymptotic settings, and are for example unable to deal with certain spatial graphs, inapplicable to small-to-medium sized social networks and incapable of capturing 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='11443v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='LG] 26 Jan 2023 the inherent multi-scale nature of molecular graphs (as further discussed below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Finally, hardly any work has been done on relating the stability to input-signal perturbations to network properties such as the interplay of utilized filters or employed non-linearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' The main focus of this work is to provide alleviation in this situation and develop a ’general theory of stability’ for GCNs – agnostic to the types of utilized filters, graph shift operators and non-linearities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' with practically relevant transferability guarantees not contingent on potentially underlying limit objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' To this end, Section 2 recapitulates the fundamentals of GCNs in a language adapted to our endeavour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Sections 3 and 4 discuss stability to node- and edge-level perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Section 5 discusses stability to structural perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Section 6 discusses feature aggregation and Section 7 provides numerical evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 2 GCNS VIA COMPLEX ANALYSIS AND OPERATOR THEORY Throughout this work, we will use the label G to denote both a graph and its associated vertex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Taking a signal processing approach, we consider signals on graphs as opposed to graph embeddings: Node-Signals: Node-signals on a graph are then functions from G to the complex numbers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' elements of C|G| (with |G| the cardinality of G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We allow nodes i P G in a given graph to have weights µi not necessarily equal to one and equip the space C|G| with an inner product according to xf, gy “ ř iPG fpiqgpiqµi to account for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We denote the hence created Hilbert space by ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Characteristic Operators: Fixing an indexing of the vertices, information about connectivity within the graph is encapsulated into the set of edge weights, collected into the adjacency matrix W and (diagonal) degree matrix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Together with the weight matrix M :“ diag ´ tµiu|G| i“1 ¯ , various standard geometry capturing characteristic operators – such as weighted adjacency matrix M ´1W, graph Laplacian ∆ :“ M ´1pD ´ Wq and normalized graph Laplacian L :“ M ´1D´ 1 2 pD ´ WqD´ 1 2 can then be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For undirected graphs, all of these operators are self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' On directed graphs, they need not even be normal (T ˚T “ TT ˚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We shall remain agnostic to the choice of characteristic operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' differentiating only between normal and general operators in our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Functional Calculus Filters: A crucial component of GCNs are functional calculus filters, which arise from applying a function g to an underlying characteristic operator T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' creating a new operator gpTq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Various methods of implementations exist, all of which agree if multiple are applicable: GENERIC FILTERS: If (and only if) T is normal, we may apply generic complex valued functions g to T: Writing normalized eigenvalue-eigenvector pairs of T as pλi, φiq|G| i“1 one defines gpTqψ “ ř|G| i“1 gpλiqxφi, ψyℓ2pGqφi for any ψ P ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' One has }gpTq}op “ supλPσpT q |gpλq|, with σpTq denoting the spectrum of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' If g is bounded, one may obtain the T-independent bound }gpTq}op ď }g}8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Keeping in mind that g being defined on all of σpTq (as opposed to all of C) is clearly sufficient, we define a space of filters which will harmonize well with our concept of transferability discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' The introduced semi-norm will quantify the stability to perturbations in coming sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Fix ω P C and C ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Define the space F cont ω,C of continuous filters on Cztω, ωu, to be the space of multilinear power-series’ gpzq “ ř8 µ,ν“0 aµν pω ´ zq´µ pω ´ zq´µ for which the semi-norm }g}F cont ω,C :“ ř8 µ,νą0 |µ ` ν|Cµ`ν´1|aµν| is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Denoting by Bϵpωq Ď C the open ball of radius ϵ around ω, one can show that for arbitrary δ ą 0 and every continuous function g defined on CzpBϵpωq Y Bϵpωqq which is regular at infinity – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' satisfies limrÑ`8 gprzq “ c P C independent of which z ‰ 0 is chosen – there is a function f P F cont ω,C so that |fpzq ´ gpzq| ď δ for all z P CzpBϵpωq Y Bϵpωqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' In other words, functions in F cont ω,C can approximate a wide class of filters to arbitrary precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' More details are presented in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ENTIRE FILTERS: If T is not necessarily normal, one might still consistently apply entire (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' everywhere complex differentiable) functions to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Detail details on the mathematical background are given in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Here we simply note that such a function g is representable as an (everywhere convergent) power series gpzq :“ ř8 k“0 ag kzk so that we may simply set gpTq “ ř8 k“0 ag k ¨ T k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For 2 the norm of the derived operator one easily finds }gpTq}op ď ř8 k“0 |ag k|}T}k op using the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' While entire filters have the advantage that they are easily and efficiently implementable – making use only of matrix multiplication and addition – they suffer from the fact that it is impossible to give a }T}op-independent bound for }gpTq}op as for continuous filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' This behaviour can be traced back to the fact that no non-constant bounded entire function exists (Bak & Newman, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' HOLOMORPHIC FILTERS: To define functional calculus filters that are both applicable to non- normal T and boundable somewhat more controlably in terms of T, one may relax the condition that g be entire to demanding that g be complex differentiable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' holomorphic) only on an open subset U Ď C of the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Here we assume that U extends to infinity in each direction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' is the complement of a closed and bounded subset of C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For any g holomor- phic on U and regular at infinity we set (with pzId ´ Tq´1 the so called reolvent of T at z) gpTq :“ gp8q ¨ Id ` 1 2πi ¿ BD gpzq ¨ pzId ´ Tq´1dz, (1) for any T whose spectrum σpTq is completely contained in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Here we have used the notation gp8q “ limrÑ`8 gprzq and taken D to an open set with nicely behaved boundary BD (more precisely a Cauchy domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We assume that D completely contains σpTq and that its closure D is completely contained in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' The orientation Figure 1: Set-Visualisations of the boundary BD is the usual positive orientation on D (such that D ’is on the left’ of BD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Using elementary facts from complex analysis it can be shown that the resulting operator gpTq in (1) is independent of the specific choice of D (Gindler, 1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' While we will present results below in terms of this general definition – remaining agnostic to numerical implementation methods for the most part – it is instructive to consider a specific exemplary setting with definite and simple numerical implementation of such filters: To this end, chose an arbitrary point ω P C and set U “ Cztωu in the definitions above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Any function g that is holomorphic on U and regular at 8 may then be represented by its Laurent series, which is of the form gpzq “ ř8 k“0 bg kpz ´ ωq´k (Bak & Newman, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For any T with σpTq Ď U (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ω R σpTq) evaluating the integral in (1) yields (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Appendix C): gpTq “ 8 ÿ k“0 bg k ¨ pT ´ ωIdq´k (2) Such filters have already been employed successfully, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' in the guise of Cayley filters (Levie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=', 2019c), which are polynomials in z`i z´i “ 1 ` 2i z´i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We collect them into a designated filter space: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For a function gpzq “ ř8 k“0 bg kpz ´ ωq´k on U :“ Cztωu define the semi-norm }g}F hol ω,C :“ ř8 k“1 |bg k|kCk´1 for C ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Denote the set of such g for which }g}F hol ω,C ă 8 by F hol ω,C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' In order to derive }T}op-independent bounds for }gpTq}op, we will need to norm-bound the resolvents appearing in (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' If T is normal, we simply have }pzId ´ Tq´1}op “ 1{distpz, σpTqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' In the general setting, following Post (2012), we call any positive function γT satisfying }pzId´Tq´1}op ď γT pzq on CzσpTq a resolvent profile of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Various methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Szehr (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' MichaelGil (2012)) to find resolvent profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Most notably Bandtlow (2004b) gives a resolvent profile solely in terms of 1{distpz, σpTqq and the departure from normality of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We then find the following result: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For holomorphic g and generic T we have }gpTq}op ď |gp8q|` 1 2π ű BD |gpzq|γT pzqd|z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Furthermore we have for any T with γT pωq ď C, that }gpTq}op ď }g}F hol ω,C as long as g P F hol ω,C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='3 (proved in Appendix D) finally bounds }gpTq}op independently of T, as long as appearing resolvents are suitably bounded;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' which – importantly – does not force }T}op to be bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Non-Linearities & Connecting Operators: To each layer of our GCN, we associate a (possibly) non-linear and Ln-Lipschitz-continuous function ρn : C Ñ C satisfying ρnp0q “ 0 which acts point-wise on signals in ℓ2pGnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' This definition allows to choose ρn “ | ¨ |, ReLu, Id or any sigmoid function shifted to preserve zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' To account for recently proposed networks where input- and ’processing’ graphs are decoupled (Alon & Yahav, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Topping et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=', 2021), and graph pooling layers (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=', 2019), we also allow signal representations in the hidden network layers n to live in 3 D OD (T) au U\\D g(T)varying graph signal spaces ℓ2pGnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Connecting operators are then (not necessarily linear) operators Pn : ℓ2pGn´1q Ñ ℓ2pGnq connecting the signal utilized of subsequent layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We assume them to be Rn-Lipschitz-continuous (}Pnpfq ´ Pnpgq}ℓ2pGn´1q ď Rn}f ´ g}ℓ2pGnqq and triviality preserving (Pnp0q “ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For our original node-signal space we also write ℓ2pGq ” ℓ2pG0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Graph Convolutional Networks: A GCN with N layers is then constructed as follows: Figure 2: Update Rule for a GCN Let us denote the width of the network at layer n by Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' The collection of hidden signals in this layer can then be thought of a single element of Ln :“ à iPKn ℓ2pGnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' (3) Further let us write the collection of functional calculus filters utilized to generate the repre- sentation of this layer by tgn ijp¨q : 1 ď j ď Kn´1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 1 ď i ď Knu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Further denoting the char- acteristic operator of this layer by Tn, the update rule (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 2) from the representation in Ln´1 to Ln is then defined on each constituent in the direct sum Ln as f n`1 i “ ρn`1 ˜ Kn ÿ j“1 gn`1 ij pTn`1qPn`1pf n j q ¸ , @1 ď i ď Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We also denote the initial signal space by Lin :“ L0 and the final one by Lout :“ LN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' The hence constructed map from the initial to the final space is denoted by Φ : Lin Ñ Lout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 3 STABILITY TO INPUT SIGNAL PERTURBATIONS In order to produce meaningful signal representations, a small input signal change should produce only a small variation in the output of our GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' This property is quantified by the Lipschitz constant of the map Φ associated to the network, which is estimated by our first result below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' With the notation of Section 2 let ΦN : Lin Ñ Lout be the map associated to an N-layer GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We have with Bn :“ b supλPσpTnq ř jPKn´1 ř iPKn |gn ijpλq|2 for all f, h P Lin that }ΦNpfq ´ ΦNphq}Lout ď ˜ N ź n“1 LnRnBn ¸ ¨ }f ´ h}Lin if Tn is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For general Tn we have for all tgiju entire, holomorphic and in F hol ω,C respectively: Bn :“ $ ’ ’ ’ ’ & ’ ’ ’ ’ % 8ř k“0 bř jPKn´1 ř iPKn |pagn ij qk|2 ¨ }Tn}k op bř jPKn´1 ř iPKn }gn ijp8q}2 ` 1 2π ű BD γT pzq bř jPKn´1 ř iPKn |gn ijpzq|2d|z| bř jPKn´1 ř iPKn }gn ij}2 F hol ω,C Appendix E contains the corresponding proof and discusses how the derived bound are not necessarily tight for sparsely connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' After Lipschitz constants of connecting operators and non- linearities are fixed, the stability constant of the network is completely controlled by the tBnu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' which for normal Tn in turn are controlled by the interplay of the utilized filters on the spectrum of Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' This allows to combine filters with supλPσpTnq |gn ijpλq| “ Op1q but supported on complimentary parts of the spectrum of Tn while still maintaining Bn “ Op1q instead of Op a Kn ¨ Kn´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' In practice one might thus penalize a ’multiple covering’ of the spectrum by more than one filter at a time during training in order to increase stability to input signal perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' If Tn is not normal but filters are holomorphic, an interplay persists – with filters now evaluated on a curve and at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 4 pn→ 2(Gn) Pn 2(Gn-1 (G l2(Gn) e2(Gn) Pn 2(Gn- 2(Gn) (G)4 STABILITY TO EDGE PERTURBATIONS Operators capturing graph-geometries might only be known approximately in real world tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' if edge weights are only known to a certain level of precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Hence it is important that graph convolutional networks be insensitive to small changes in the characteristic operators tTnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Since we consider graphs with arbitrary vertex weights tµgugPG, we also have to consider the possibility that these weights are only known to a certain level of precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' In this case, not only do the characteristic operators Tn, rTn differ, but also the the spaces ℓ2pGq, ℓ2p rGq on which they act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' To capture this setting mathematically, we assume in this section that there is a linear operator J : ℓ2pGq Ñ ℓ2p rGq facilitating contact between signal spaces (of not-necessarily the same dimension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We then measure closeness of characteristic operators in the respective spaces by considering the generalized norm- difference }pJT ´ rTJq};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' with J translating between the respective spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Before investigating the stability of entire networks we first comment on single-filter stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For normal operators we then find the following result, proved in Appendix A building on ideas first developed in (Wihler, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Denote by } ¨ }F the Frobenius norm and let T and rT be normal on ℓ2pGq and ℓ2p rGq respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Let g be Lipschitz continuous with Lipschitz constant Dg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For any linear J : ℓ2pGq Ñ ℓ2p rGq we have }gp rTqJ ´ JgpTq}F ď Dg} rTJ ´ JT}F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Unfortunately, scalar Lipschitz continuity only directly translates to operator functions if they are applied to normal operators and when using Frobenius norm (as opposed to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' spectral norm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For general operators we have the following somewhat weaker result, proved in Appendix F: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Let T, rT be operators on on ℓ2pGq , ℓ2p rGq with }T}op, } rT}op ď C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Let J : ℓ2pGq Ñ ℓ2p rGq be linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' With Kg “ 1 2π ű BD 1 |z|γT pzqγ rT pzq|gpzq|d|z| for g holomorphic and Kg “ ř8 k“1 |ag k|kCk´1 for g entire, we have }gpTqJ ´ Jgp rTq}op ď Kg ¨ }JT ´ rTJ}op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Each Kg itself is interpretable as a semi-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For GCNs we find the following (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Appendix F): Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Let ΦN, rΦN be the maps associated to N-layer graph convolutional networks with the same non-linearities and filters, but based on different graph signal spaces ℓ2pGq, ℓ2p rGq, characteristic operators Tn, rTn and connecting operators Pn, rPn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Assume Bn, rBn ď B as well as Rn, rRn ď R and Ln ď L for some B, R, L ą 0 and all n ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Assume that there are identification operators Jn : ℓ2pGnq Ñ ℓ2p rGnq (0 ď n ď N) commuting with non-linearities and connecting operators in the sense of } rPnJn´1f ´ JnPnf}ℓ2p r Gnq “ 0 and }ρnpJnfq ´ Jnρnpfq}ℓ2p r Gnq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Depending on whether normal or arbitrary characteristic operators are used, define D2 n :“ ř jPKn´1 ř iPKn D2 gn ij or D2 n :“ ř jPKn´1 ř iPKn K2 gn ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Choose D such that Dn ď D for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Finally assume that }JnTn ´ rTnJn}˚ ď δ and with ˚ “ F if both operators are normal and ˚ “ op otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Then we have for all f P Lin and with Jn the operator that the Kn copies of Jn induce through concatenation that }rΦpJ0fq ´ JNΦpfq} Ă Lout ď N ¨ DRL ¨ pBRLqN´1 ¨ }f}Lin ¨ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' The result persists with slightly altered constants, if identification operators only almost commute with non-linearities and/or connecting operators, as Appendix G further elucidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Since we estimated various constants (Bn, Dn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=') of the individual layers by global ones, the derived stability constant is clearly not tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' However it portrays requirements for stability to edge level perturbations well: While the (spectral) interplay of Section 3 remains important, it is now especially large single-filter stability constants in the sense of Lemmata 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2 that should be penalized during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 5 STABILITY TO STRUCTURAL PERTURBATIONS: TRANSFERABILITY While the demand that } rTJ ´ JT} be small in some norm is well adapted to capture some notions of closeness of graphs and characteristic operators, it is too stringent to capture others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' As an illustrative example, further developed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2 and numerically investigated in Section 7 below, suppose we are given a connected undirected graph with all edge weights of order Op1{δq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' With the Laplacian as characteristic operator (governing heat-flow in Physics (Cole, 2011)), we may think of this graph as modelling an array of coupled heat reservoirs with edge weights corresponding to 5 heat-conductivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' As 1{δ Ñ 8, the conductivities between respective nodes tend to infinity, heat exchange is instantaneous and all nodes act as if they are fused together into a single large entity – with the graph together with its characteristic operator behaving as an effective one-dimensional system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' This ’convergent’ behaviour is however not reflected in our characteristic operator, the graph Laplacian ∆δ: Clearly }∆δ}op “ 1{δ ¨ }∆1}op Ñ 8 as 1{δ Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Moreover, we would also expect a Cauchy- like behaviour from a ’convergent system’, in the sense that if we for example keep 1{δa ´ 1{δb “ 1 constant but let p1{δaq, p1{δbq Ñ 8 we would expect }∆δa ´ ∆δb}op Ñ 0 by a triangle-inequality argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' However, we clearly have }∆δa ´∆δb}op “ |1{δa ´1{δb|¨}∆1}op “ }∆1}op, which does not decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' The situation is different however, when considering resolvents of the graph Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' An easy calculation (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Appendix H) yields }pωId ´ ∆δbq´1 ´ pωId ´ ∆δaq´1}op “ Opδa ¨ δbq so that we recover the expected Cauchy behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' What is more, we also find the convergence pωId ´ ∆δq´1 Ñ P0 ¨ pω ´ 0q´1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' where P0 denotes the projection onto the one-dimensional lowest lying eigenspace of the ∆δs (spanned by the vectors with constant entries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We may interpret pω´0q´1 as the resolvent of the graph Laplacian of a singleton (since such a Laplacian is identically zero) and thus now indeed find our physical intuition about convergence to a one-dimensional system reflected in our formulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Motivated by this example, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 develops a general theory for the difference in outputs of networks evaluated on graphs for which the resolvents Rω :“ pωId ´ Tq´1 and rRω :“ pωId ´ rTq´1 of the respective characteristic operators are close in some sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Subsequently, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2 then further develops our initial example while also considering an additional setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 GENERAL THEORY Throughout this section we fix a complex number ω P C and for each operator T assume ω, ω R σpTq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' This is always true for ω with |ω| ě }T}op, but if T is additionally self adjoint one could set ω “ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' If T is non-negative one might choose ω “ p´1q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' As a first step, we then note that the conclusion of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 can always be satisfied if we chose J ” 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' To exclude this case – where the application of J corresponds to losing too much information – we follow Post (2012) in making the following definition: Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Let J : ℓ2pGq Ñ ℓ2p rGq and rJ : ℓ2p rGq Ñ ℓ2pGq be linear, and let T ( rT) be operators on (ℓ2pGq) (ℓ2p rGq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We say that J and rJ are ϵ-quasi-unitary with respect to T, rT and ω if }Jf}ℓ2p r Gq ď 2}f}ℓ2pGq, }pJ ´ rJ˚qf}ℓ2p r Gq ď ϵ}f}ℓ2pGq, }pId ´ rJJqRωf}ℓ2pGq ď ϵ}f}ℓ2pGq, }pId ´ J rJq rRωu}ℓ2p r Gq ď ϵ}u}ℓ2p r Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' (4) The motivation to include the resolvents in the norm estimates (4) comes from the setting where T “ ∆ is the graph Laplacian and ω “ p´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' In that case, the left equation in (4 is for example automatically fulfilled when demanding }pId ´ rJJqf}2 ℓ2pGq ď ϵp}f}2 ` E∆pfqq 1 2 , with E∆p¨q “ x¨, ∆¨yℓ2pGq the (positive) energy form induced by the Laplacian ∆ (Post, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' This can thus be interpreted as a relaxation of the standard demand }pId ´ rJJq}op ď ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Relaxing the demands of Section 4, we now demand closeness of resolvents instead of closeness of operators: Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' If, for ω P C and linear J : ℓ2pGq Ñ ℓ2p rGq the resolvents Rω and rRω satisfy }p rRωJ ´ JRωqf}ℓ2p r Gq ď ϵ}f}ℓ2pGq for all f P ℓ2pGq, T and rT are called ω-ϵ-close with identifica- tion operator J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' If additonally }p rR˚ ωJ ´ JR˚ ωqf}ℓ2p r Gq ď ϵ}f}ℓ2pGq, they are doubly ω-ϵ-close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Our first result establishes that operators being (doubly-)ω-ϵ-close indeed has useful consequences: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Let T ( rT) be operators on ℓ2pGq (ℓ2p rGq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' If these operators are ω-ϵ-close with identification operator J, and }Rω}op, } rRω}op ď C we have }JgpTq ´ gp rTqJ}op ď Kg ¨ }p rRωJ ´ JRωq}op with Kg “ 1 2π ű BDp1 ` |z ´ ω|γT pzqqp1 ` |z ´ ω|γ rT pzqq|gpzq|d|z| for holomorphic g, Kg “ }g}F hol ω,C if g P F hol ω,C and Kg “ }g}F cont ω,C for T, rT normal and doubly ω-ϵ-close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' This result may then be extended to entire networks, as detailed in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='4 below whose statement persists with slightly altered stability constants, if identification operators only almost commute with non-linearities and/or connecting operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Proofs are contained in Appendix I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Let ΦN, rΦN be the maps associated to N-layer graph convolutional networks with the same non-linearities and functional calculus filters, but based on different graph signal spaces 6 ℓ2pGnq, ℓ2p rGnq, characteristic operators Tn, rTn and connecting operators Pn, rPn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Assume Bn, rBn ď B as well as Rn, rRn ď R and Ln ď L for some B, R, L ą 0 and all n ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Assume that there are identification operators Jn : ℓ2pGnq Ñ ℓ2p rGnq (0 ď n ď N) commuting with non- linearities and connecting operators in the sense of } rPnJn´1f ´JnPnf}ℓ2p r Gnq “ 0 and }ρnpJnfq´ Jnρnpfq}ℓ2p r Gnq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' define D2 n :“ ř jPKn´1 ř iPKn K2 gn ij with Kgn ij as in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Choose D such that Dn ď D for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Finally assume that }JnpωId ´ Tnq´1 ´ pωId ´ rTnq´1Jn}op ď ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' If filters in F cont ω,C are used, assume additionally that }JnppωId´Tnq´1q˚´ppωId´ rTnq´1q˚Jn}op ď ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Then we have for all f P Lin and with Jn the operator that the Kn copies of Jn induce through concatenation that }rΦNpJ0fq ´ JNΦNpfq} Ă Lout ď N ¨ DRL ¨ pBRLqN´1 ¨ }f}Lin ¨ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2 EXEMPLARY APPLICATIONS Collapsing Strong Edges: We first pick our example from the beginning of section 5 up again and generalize it significantly: We now consider the graph that we collapse to a single node to be a sub-graph (of strong edges) embedded into a larger graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Apart from coupled heat reservoirs, this setting also e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' captures the grouping of close knit communities within social networks into single entities, the scale-transition of changing the description of (the graph of) a molecule from individual atoms interacting via the coulomb potential Z1Z2{R (with R the distance and Z1, Z2 atomic charges) to the interaction of (functional) groups comprised of closely co-located atoms, or spatial networks if weights are set to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' inverse distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' In what follows, we shall consider two graphs with vertex sets G and rG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We consider G to be a subset of the vertex set rG and think of the graph corresponding to G as arising in a collapsing procedure from the ’larger’ graph rG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' More precisely, we assume that the vertex set rG can be split into three disjoint subsets rG “ rGLatin Ť rGGreek Ťt‹u (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We assume that the adjacency matrix Ă W when restricted to Latin vertices or a Latin vertex and the exceptional node ’‹’ is of order unity p Ą Wab, Ă Wa‹ “ Op1q, @a, b P rGLatinq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For Greek indices, we assume that we may write Ă Wαβ “ ωαβ δ and Ă Wα‹ “ ωα‹ δ such that pωαβ, ωα‹ “ Op1q for all α, β P rGGreek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We also assume that the sub-graph corresponding to vertices in rGGreek Ťt‹u is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We then take G “ rGLatin Ťt‹u (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' again Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' The adjacency ma- trix W on this graph is constructed by defining Wab “ Ă Wab, @a, b P rGLatin and setting (with Wa‹ ” W‹a) W‹a :“ Ă Wa‹ ` ÿ βP r GGreek Ă Waβ ´ @a P rGLatin ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We also allow our graph rG to posses node-weights trµrgurgP r G that are not necessarily equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' The Laplace operator ∆ r G acting on the graph signal space ℓ2p rGq induces a positive semi-definite and convex Figure 3: Collapsed (left) and original (right) Graphs energy form on this signal space via E r Gpuq :“ xu, ∆ r Guyℓ2p r Gq “ ř g,hP r G Ă Wgh|upgq´uphq|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Using this energy form, we now define a set comprised of |G| signals, all of which live in ℓ2p rGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' These signals are used to facilitate contact between the respective graph signal spaces ℓ2pGq and ℓ2p rGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For each g P G, define the signal ψδ g P ℓ2p rGq as the unique solution to the convex optimization program min E r Gpuq subject to uphq “ δhg for all h P rGLatin ď t‹u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' (5) Given the boundary conditions, what is left to determine in the above optimization program are the ’Greek entries’ ψδ gpαq of each ψδ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' As Appendix J further elucidates, these can be calculated explicitly and purely in terms of the inverse of ∆ r G restricted to Greek indices as well as (sub-)columns of the adjacency matrix Ă W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Node-weights on G are then defined as µδ g :“ ř hP r G ψδ gphq ¨ rµh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We denote 7 the corresponding signal space by ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Importantly, one has µδ a Ñ rµa for any Latin index and µδ ‹ Ñ rµ‹ ` ř αP r GGreek rµα as δ Ñ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' which recovers our physical intuition about heat reservoirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' To translate signals from ℓ2pGq to ℓ2p rGq and back, we define two identification operators J : ℓ2pGq Ñ ℓ2p rGq and rJ : ℓ2p rGq Ñ ℓ2pGq via Jf :“ ř gPG fpgq ¨ ψδ g and p rJuqpgq :“ xu, ψδ gyℓ2p r Gq{µδ g for all f P ℓ2pGq, u P ℓ2p rGq and g P G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Our main theorem then states the following: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' With definitions and notation as above, there are constants K1, K2 ě 0 such that the operators J and rJ are pK1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' δq-quasi-unitary with respect to ∆ r G, ∆G and ω “ p´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Furthermore, the operators ∆ r G and ∆G are p´1q-pK2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' δq close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' with identification operator J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Appendix J presents the (fairly involved) proof of this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Importantly, the size of the constants K1, K2 is independent of the cardinality (or more precisely the total weight) of rGLatin, implying that Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='6 also remains applicable in the realm of large graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Finally we note, that this stability result is contingent on the use of the (un-normalized) graph Laplacian (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Appendix K): Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' In the setting of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='6 denote by T ( rT) adjacency matrices or normalized graph Laplacians on ℓ2pGq (ℓ2pGq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' There are no functions η1, η2 : r0, 1s Ñ Rě0 with ηipδq Ñ 0 as δ Ñ 0 (i “ 1, 2), families of identification operators Jδ, rJδ and ω P C so that Jδ and rJδ are η1pδq-quasi-unitary with respect to rT, T and ω while the operators rT and T remain ω-η2pδq close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' The Realm of Large Graphs: In order to relate our transferability framework to the literature, we consider an ’increasing’ sequence of graphs (Gn Ď Gn`1) approximating a limit object, so that the transferability framework of Levie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' (2019a) is also applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We choose the limit object to be the circle of circumference 2π and our approximating graphs to be the closed path-graph on N vertices Figure 4: Closed Path-Graphs equidistantly embedded into the circle (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Fig 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' With h “ 2π{N the node-distance, we set weights to 1{h2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ensuring consistency with the ’continuous’ Laplacian in the limit N Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' More details are presented in Appendix L, which also contains the proof of the corresponding transferability result: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' In the above setting choose all node-weights equal to one and N to be odd for definiteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' There exists constants K1, K2 “ Op1q so that for each N ě 1, there exist identification operators J, rJ mapping between ℓ2pGNq and ℓ2pGN`1q so that J and rJ are pK1{Nq-quasi-unitary with respect to ∆GN , ∆GN`1 and ω “ p´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Furthermore, the operators ∆GN and ∆GN`1 are p´1q-pK2{Nq close with identification operator J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='3 then implies an Op 1 N q-decay of }gpTqJ ´ Jgp rTq}op for fixed g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' This reduces to an Op ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' N N q-decay for Levie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' (2019a) (ibid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Theorem 5, pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 3) assuming a similar decay of operator- distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Our framework might this capture transferability properties other approaches could miss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 6 GRAPH LEVEL STABILITY To solve tasks such as graph classification or regression over multiple graphs, graphs of varying sizes need to be represented in a common feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Here we show that aggregating node-level features into such graph level features via p-norms (}f}ℓppGq :“ př gPG |fg|pµgq1{p) preserves stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' To Figure 5: Graph Level Aggregation this end, let Lout be a target space of a GCN in the sense of (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' On each of the (in total Kout) ℓ2pGoutq summands of Lout, we may apply the map fi ÞÑ }fi}ℓppGoutq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Stacking these maps, we build a map from Lout to RKout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Concatenating the map ΦN associated to an N-layer GCN with this map yields a map from Lin to RKout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We denote it by Ψp N and find: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For p ě 2 we have in the setting of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 that }Ψp Npfq ´ Ψp Nphq}RKout ď ´śN n“1 LnRnBn ¯ ¨ }f ´ h}Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' In the setting of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='3 or 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='4 and under the additional assumption that the ’final’ identification operator JN satisfies ˇˇ}JNfi}ℓkp r GNq ´ }fi}ℓkpGNq ˇˇ ď δ ¨ K ¨ }fi}ℓ2pGNq for all fi P ℓ2pGNq, we have }Ψp Npfq ´ rΨp NpJ0fq}RKout ď pN ¨ DRL ` K ¨ pBRLqq ¨ pBRLqN´1 ¨ }f}Lin ¨ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 8 /· Ilep(Gout) R l2(Gin)— .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ep(Gout) C→l2 (Gout) R l2(Gin)-N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ep(Gout) →l2(Gout) RDerived stability results thus persist (under mild assumptions) if graph level features are aggregated via p-norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Appendix M contains the corresponding proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 7 NUMERICAL RESULTS We focus on investigating structural perturbations, as correspond- ing results are most involved and novel: We first consider a graph on 5 nodes with an adjacency matrix A with Op1q-entries (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 30 in Appendix N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We then scale A by 1{δa and 1{δb (with 1 δa ´ 1 δb “ 1) respectively and consider the norm-difference between associated Laplacians and resolvents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 6 (a) then illustrate the theoretical result (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Section 5) that resolvent- instead of Laplacian-differences capture the conver- gence behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Embedding the considered graph into a larger graph (Ă W P R8ˆ8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' (31) in Appendix N), we consider the collapsing edge setting of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 6 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' As expected, the corresponding resolvents do approach each other as δ Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Contrary to the theoretical bound in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='3, differences of resolvent-monomials decrease as their power k increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Beyond small graphs – inaccessible to traditional asymptotic methods – our method is also applicable to the large-graph setting: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 7 picks up the example of an ’increasing’ graph sequence ’approximating’ the circle again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' As predicted in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2, the difference in resolvents decays (9 1 N ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 10 in Appendix N shows how the difference in Laplacians diverges instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Hence Figure 6: Edge-Collapse Stability Figure 7: The Large-N Regime our framework might capture stability properties traditional ap- proaches could miss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Finally, we investigate the transferability of a two-layer GCN with 16 nodes per hidden Layer combined with the aggregation method of Section 6 into a graph-level map Ψp 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Filters are of the form (2) up to order k “ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Coefficients tbg ku are sampled uniformly from r´100, 100s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Feature vectors are generated on the QM7 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' There each graph represents a molecule;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' nodes correspond to individual atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Adjacency matrices are given by Ă Wij “ ZiZj{}xi ´ xj} with Zi (xi) the atomic charge (equilib- rium position) of atom i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We choose node-weights as rµi “ Zi and the Laplacian as characteristic operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Leading up to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 8 we consider the graph of methane (5 Nodes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' one Carbon (Z1 “ 6) and four Hydrogen nodes (Zią1 “ 1)) and deflect one of the Hy- drogen atoms (i “ 2) out of equilibrium and along a straight line towards the Carbon atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We then consider the transferability of the entire GCN between the resulting graph and an effective graph combining Carbon and deflected Hydrogen into a single node "‹" with weight µ‹ “ Z1 ` Z2 “ 7 located at the equilibrium position of Carbon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' With J translating from effective to original description, we consider }Ψp 2pfq ´ Ψp 2pJfq}R16 (averaged over 100 random unit-norm choices of f) as a function of }x1 ´x2}´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' At equilibrium the transferability error is Op1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' It decreases fast with decreasing Carbon-Hydrogen distance, with the choice of Figure 8: GCN Transferability representation (effective vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' original) quickly becoming insignificant for generated feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 8 DISCUSSION A theoretically well founded framework capturing stability properties of GCNs was developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We related node-level stability to (spectral) covering properties and edge-level stability to introduced semi-norms of employed filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For non-normal characteristic operators, tools from complex analysis provided grounds for derived stability properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We introduced a new notion of stability to structural 9 102 II sa -△s ll op 101 100 10-1 ResolventDifferences Operator Differences 10-2 10-3 R-1(s）)- R-1(s)lop 10-4 10-5 0 20 40 1/8a 60 80 100 a k=1 10-2 IlRk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='J - JRillop k=2 k=3 10-3 k=4 k=5 10-4 10-5 10-6 10-7 10-8 10-9 (b) 0 20 40 1/8 60 80 10010-2 10-3 Resolvent Differences R-1(△GN+1)J- JR-1(△G)lop 10-4 10-5 10-6 10-7 10-8 0 250 500 750 1000 1250 1500 1750 2000 N蚂(Jf)－() p= 2 R16 p = 3 p= 4 100 p = 5 p = 6 p = 7 p-Norm Differences 10-1 10-2 10-3 0 20 40 60 80 100 equilibrium distancel /-1 C1perturbations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' highlighted the importance of the resolvent and detailed how the developed line of thought captures relevant settings of structural changes such as the collapse of a strongly connected sub-graph to a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' There – precisely if the graph Laplacian was employed – the transferability error could be bounded in terms of the inverse characteristic coupling strength on the sub-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' REFERENCES Uri Alon and Eran Yahav.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='21413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Sohir Maskey, Ron Levie, and Gitta Kutyniok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Transferability of graph neural networks: an extended graphon approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' CoRR, abs/2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='10096, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' URL https://arxiv.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Norm estimates for resolvents of non-selfadjoint operators having hilbert-schmidt inverse ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Olaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Spectral Analysis on Graph-like Spaces / by Olaf Post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Lecture Notes in Mathematics, 2039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Springer Berlin Heidelberg, Berlin, Heidelberg, 1st ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' edition, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ISBN 3-642-23840-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Olaf Post and Jan Simmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Approximation of fractals by discrete graphs: norm resolvent and spectral convergence, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' URL https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='org/abs/1704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='00064.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Luana Ruiz, Luiz F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Chamon, and Alejandro Ribeiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Graphon neural networks and the trans- ferability of graph neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' In Hugo Larochelle, Marc’Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan-Tien Lin (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ), Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' URL https://proceedings.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Müller, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' von Lilienfeld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Fast and accurate modeling of molecular atomization energies with machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Physical Review Letters, 108:058301, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Oleg Szehr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Eigenvalue estimates for the resolvent of a non-normal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Journal of Spectral Theory, 4(4):783–813, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='4171/jst/86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' URL https://doi.' metadata={'source': 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https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='org/abs/2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='14522.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Thomas Wiatowski and Helmut Bölcskei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' A mathematical theory of deep convolutional neural networks for feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' IEEE Transactions on Information Theory, 64:1845–1866, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Wihler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' On the hölder continuity of matrix functions for normal matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Journal of inequalities in pure and applied mathematics, 10(4), Dec 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ISSN 1443-5756.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' emis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='de/journals/JIPAM/images/276_09_JIPAM/276_09_www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 11 A SOME CONCEPTS IN LINEAR ALGEBRA In the interest of self-containedness, we provide a brief review of some concepts from linear algebra utilized in this work that might potentially be considered more advanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Presented results are all standard;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' a very thorough reference is Michael Reed (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Hilbert Spaces: To us, a Hilbert space — often denoted by H — is a vector space over the complex numbers which also has an inner product — often denoted by x¨, ¨yH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Prototypical examples are given by the Euclidean spaces Cd with inner product xx, yyCd :“ řd i“1 xiyi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Associated to an inner product is a norm, denoted by } ¨ }H and defined by }x}H :“ a xx, xyH for x P H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Direct Sums of Spaces: Given two potentially different Hilbert spaces H and p H, one can form their direct sum H ‘ p H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Elements of H ‘ p H are vectors of the form pa, bq, with a P H and b P p H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Addition and scalar multiplication are defined in the obvious way by pa, bq ` λpc, dq :“ pa ` λc, b ` λdq for a, c P H, b, d P p H and λ P C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' The inner product on the direct sum is defined by xpa, bq, pc, dqyH‘ p H :“ xa, cyH ` xb, dy p H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' As is readily checked, this implies that the norm } ¨ }H‘ p H on the direct sum is given by }pa, bq}2 H‘ p H :“ }a}2 H ` }b}2 p H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Standard examples of direct sums are again the Euclidean spaces, where one has Cd “ Cn ‘ Cm if m ` n “ d, as is easily checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' One might also consider direct sums with more than two summands, writing Cd “ ‘d i“1C for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' In fact, one might also consider infinite sums of Hilbert spaces: The space ‘8 i“1Hi is made up of those elements a “ pa1, a2, a3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='q with ai P Hi for which the norm }a}2 ‘8 i“1Hi :“ 8 ÿ i“1 }ai}2 Hi is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' This means for example that the vector p1, 0, 0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='q is in ‘8 i“1C, while p1, 1, 1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='q is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Direct Sums of Maps: Suppose we have two collections of Hilbert spaces tHiuΓ i“1, t r HiuΓ i“1 with Γ P N or Γ “ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Suppose further that for each i ď Γ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' i ă Γ) we have a (not necessarily linear) map Ji : Hi Ñ r Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Then the collection tJiuΓ i“1 of these ’component’ maps induce a ’composite’ map J : ‘Γ i“1Hi ÝÑ ‘Γ i“1 r Hi between the direct sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Its value on an element a “ pa1, a2, a3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='q P ‘Γ i“1Hi is defined by J paq “ pJ1pa1q, J2pa2q, J3pa3q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='q P ‘Γ i“1 r Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Strictly speaking, one has to be a bit more careful in the case where Γ “ 8 to ensure that }J paq}‘8 i“1 r Hi ‰ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' This can however be ensured if we have }Jipaiq} r Hi ď C}ai}Hi for all 1 ď i and some C independent of all i, since then }J paq}‘8 i“1 r Hi ď C}a}‘8 i“1Hi ď 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' If each Ji is a linear operator, such a C exists precisely if the operator norms (defined below) of all Ji are smaller than some constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Operator Norm: Let J : H Ñ r H be a linear operator between Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We measure its ’size’ by what is called the operator norm, denoted by } ¨ }op and defined by }J}op :“ sup ψPH,}ψ}H“1 }Aψ} r H }ψ}H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 12 Adjoint Operators Let J : H Ñ r H be a linear operator from the Hilbert space H to the Hilbert space r H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Its adjoint J˚ : r H Ñ H is an operator mapping in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' It is uniquely determined by demanding that xJf, uy r H “ xf, J˚uyH holds true for arbitrary f P H and u P r H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Normal Operators: If a linear operator ∆ : H Ñ H maps from and to the same Hilbert space, we can compare it directly with its adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' If ∆∆˚ “ ∆˚∆, we say that the operator ∆ is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Special instances of normal operators are self-adjoint operators, for which we have the stronger property ∆ “ ∆˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' If an operator is normal, there are unitary maps U : H Ñ H diagonalizing ∆ as U ˚∆U “ diagpλ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='λnq, with eigenvalues in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We call the collection of eigenvalues the spectrum σp∆q of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' If dim H “ d, we may write σp∆q “ tλud i“1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' It is a standard exercise to verify that each eigenvalue satisfies |λi| ď }∆}op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Associated to each eigenvalue is an eigenvector φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' The collection of all (normalized) eigenvectors forms an orthonormal basis of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We may then write ∆f “ dÿ i“1 λi xφi, fyHφi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Resolvent of an Operator: Given an operator T on some Hilbert space H, we have by definition that the operator pT ´ zq : H Ñ H is invertible precisely if z ‰ σpTq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' In this case we write RzpTq “ pzId ´ Tq´1 and call this operator the resolvent of T at z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' If T is normal it can be proved that the norm of the resolvent satisfies }RzpTq}op “ 1 distpz, σp∆qq, where distpz, σp∆qq denotes the minimal distance between z and any eigenvalue of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For non- normal operators, one can prove }RzpTq}op ď γT pzq with γT pzq “ exp r2}T}1{dpz, σpTqqs {dpz, σpTqq as is proved in Bandtlow (2004a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Frobenius Norm: Given two finite dimensional Hilbert spaces H1 and H2 with orthonormal bases tφ1 i ud1 i“1 and tφ1 i ud1 i“1, the Frobenius norm } ¨ }F of an operator A : H1 Ñ H2 may be defined as }A}2 2 :“ d2 ÿ i“1 d1 ÿ j“1 |Aij|2 with Aij the matrix representation of A with respect to the bases tφ1 i ud1 i“1 and tφ1 i ud1 i“1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' It is a standard exercise to verify that this norm is indeed independent of any choice of basis and hence invariant under multiplying A with a unitary on either the left or the right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' More precisely, if U : H2 Ñ H2 and V : H1 Ñ H1 are unitary, we have }UAV }2 F “ }A}2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Frobenius norms can be used to transfer Lipschitz continuity properties of complex functions to the setting of functions applied to normal operators: Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Let g : C Ñ C be Lipschitz continuous with Lipschitz constant Dg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' This implies }gpXqJ ´ JgpY q}F ď Dg ¨ }X ´ Y }F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' for normal operators X on H2, Y on H1 and any linear map J : H1 Ñ H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' This proof is a modified version of the proof in Wihler (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Let U, W be unitary (with respect to the inner product x¨, ¨yH) operators diagonalizing the normal operators X and Y as V ˚XV “ diagpλ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='λd2q “: DpXq W ˚Y W “ diagpµ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='µd1q “: DpY q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Since the Frobenius norm is invariant under unitary transformations we find }gpXqJ ´ JgpY q||2 F “ ||gpV DpXqV ˚q ´ gpWDpY qW ˚q}2 F “ }V gpDpXqqV ˚J ´ JWgpDpY qqW ˚}2 F “ }gpDpXqqV ˚JW ´ V ˚JWgpDpY qq}2 F “ ÿ i,j |pgpDpXqqV ˚JW ´ V ˚JWgpDpY qqqij|2 “ ÿ i,j ˇˇˇˇˇ ÿ k rgpDpXqqsikrV ˚JWskj ´ rV ˚JWsikrgpDpY qqskj ˇˇˇˇˇ 2 “ ÿ i,j |rV ˚Wsij|2 |gpλjq ´ gpµiq|2 ď ÿ i,j |rV ˚Wsij|2 D2 g|λj ´ µi|2 “ D2 g}X ´ Y }2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' B APPROXIMATING BOUNDED CONTINUOUS FILTERS Let us recall Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1: Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Fix ω P C and C ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Define the space F cont ω,C of continuous filters on Cztω, ωu, to be the space of multilinear power-series’ gpzq “ ř8 µ,ν“0 aµν pω ´ zq´µ pω ´ zq´µ for which the norm }g}F cont ω,C :“ ř8 µ,ν“0 |µ ` ν|Cµ`ν|aµν| is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We now prove that upon denoting by Bϵpωq Ď C the open ball of radius ϵ around ω, one can show that for arbitrary δ ą 0 and every continuous function g defined on CzpBϵpωq Y Bϵpωqq which is regular at infinity – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' satisfies limrÑ`8 gprzq “ c P C independent of which z ‰ 0 is chosen – there is a function f P F cont ω,C so that |fpzq ´ gpzq| ď δ for all z P CzpBϵpωq Y Bϵpωqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Making use of the Stone-Weierstrass theorem for complex functions, it suffices to prove that for every point z in CzpBϵpωq Y Bϵpωqq there are functions f and g in F cont ω,C for which fpzq ‰ gpzq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' But this is obvious since pω ´ zq´1 is injective on CzpBϵpωq Y Bϵpωqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' C COMPLEX ANALYSIS A general reference for topics discussed in this section is Bak & Newman (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For a complex valued function f of a single complex variable, the derivative of f at a point z0 P C in its domain of definition is defined as the limit f 1pz0q :“ lim zÑz0 fpzq ´ fpz0q z ´ z0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For this limit to exist, it needs to be independent of the ’direction’ in which z approaches z0, which is a stronger requirement than being real-differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' A function is called holomorphic on an open set U if it is complex differentiable at every point in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' It is called entire if it is complex differentiable at every point in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Every entire function has an everywhere convergent power series representation gpzq “ 8 ÿ k“0 agzk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' (6) 14 If a function g is analytic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' can be expanded into a power series), we have gpλq “ ´ 1 2πi ¿ S gpzq λ ´ z dz (7) for any circle S Ď C encircling λ by Cauchy’s integral formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' In fact, the integration contour need not be a circle S, but may be the boundary of any so called Cauchy domain containing λ: Definition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' A subset D of the complex plane C is called a Cauchy domain if D is open, has a finite number of components (the closure of two of which are disjoint) and the boundary of BD of D is composed of a finite number of closed rectifiable Jordan curves, no two of which intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Equation (7) forms the backbone of complex analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Since the integral I :“ ´ 1 2πi ¿ BD gpzqpzId ´ Tq´1dz (8) is well defined for holomorphic gp¨q and any operator T for which σpTq and BD are disjoint (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Post (2012) for details), we can essentially take (8) as a defining equation through which one might apply holomorphic functions to operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' While functions that are everywhere complex differentiable have a series representation according to (6), complex functions that are holomorphic only on Cztωu have a series representation (called Laurent series) according to gpzq “ 8 ÿ k“´8 akpz ´ ωqk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' If these functions are assumed to be regular at infinity, no terms with positive exponent are permitted and (changing the indexing) we may thus write gpzq “ 8 ÿ k“0 akpz ´ ωq´k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Motivated by this, we now prove the following consistency result: Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' With the notation of Section 2 we have for any k ě 1 and ω R σpTq that pω ¨ Id ´ Tq´k :“ 1 2πi ¿ BD pω ´ zq´k ¨ pzId ´ Tq´1dz, where we interpret the left hand side of the equation in terms of inversion and matrix powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We first note that we may write RλpTq “ 8 ÿ n“0 pλ ´ ωqnp´1qnRωptqn`1 for |λ ´ ω| ď }RωpTq} using standard results in matrix analysis (namely the ’Neumann Characteri- sation of the Resolvent’ which is obtained by repeated application of a resolvent identity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Post (2012) for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We thus find 1 2πi ¿ BD ˆ 1 ω ´ z ˙k 1 zId ´ T dz “ 1 2πi ¿ BD ˆ 1 ω ´ z ˙k 8 ÿ n“0 pω ´ zqnRωpTqn`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Using the fact that 1 2πi ¿ BD pz ´ ωqn´k´1dz “ δnk then yields the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 15 D PROOF OF LEMMA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='3 We want to prove the following: Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For holomorphic g and generic T we have }gpTq}op ď |gp8q| ` 1 2π ű BD |gpzq|γT pzqd|z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Furthermore we have for any T with γT pωq ď C, that }gpTq}op ď }g}F hol ω,C as long as g P FC,ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We first note ›››››› gp8q ¨ Id ` 1 2πi ¿ BD gpzq ¨ pzId ´ Tq´1dz ›››››› op ď }gp8q ¨ Id}op ` ›››››› 1 2πi ¿ BD gpzq ¨ pzId ´ Tq´1dz ›››››› op ď |gp8q| ` 1 2π ¿ BD |gpzq| ››¨pzId ´ Tq´1›› op d|z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' The first claim thus follows together with }RzpTq}op ď γT pzq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' The second claim can be derived as follows: }gpTq}op “ ››››› 8 ÿ k“0 bg kpT ´ ωq´k ››››› op ď 8 ÿ k“0 |bg k| ››pT ´ ωq´k›› op ď 8 ÿ k“0 |bg k|γT pωqk ď 8 ÿ k“0 |bg k|Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' E PROOF OF THEOREM 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 AND TIGHTNESS OF RESULTS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We want to prove the following: Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' With the notation of Section 2 let ΦN : Lin Ñ Lout be the map associated to an N-layer GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We have }ΦNpfq ´ ΦNphq}Lout ď ˜ N ź n“1 LnRnBn ¸ ¨ }f ´ h}Lin with Bn :“ b supλPσpTnq ř jPKn´1 ř iPKn |gn ijpλq|2 if Tn is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For general Tn we have for all tgiju entire, holomorphic and in Fω,C respectively: Bn :“ $ ’ ’ ’ ’ & ’ ’ ’ ’ % 8ř k“0 bř jPKn´1 ř iPKn |pagn ij qk|2 ¨ }Tn}k op bř jPKn´1 ř iPKn }gn ijp8q}2 ` 1 2π ű Γ γT pzq bř jPKn´1 ř iPKn |gn ijpzq|2d|z| bř jPKn´1 ř iPKn }gn ij}2 ω,C Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Given input signals f, hn P Lin, let us – sticking to the notation introduced in Section 2 – denote the intermediate signal representations in the intermediate layers Ln by f n, hn P Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' With the update rule described in Section 2 and the norm induced on each Ln as described in Appendix A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' we then have }f n`1 ´ hn`1}2 Ln`1 “ Kn`1 ÿ i“1 ›››››ρn`1 ˜ Kn ÿ j“1 gn`1 ij pTn`1qPn`1pf n j q ¸ ´ ρn`1 ˜ Kn ÿ j“1 gn`1 ij pTn`1qPn`1phn j q ¸››››› 2 ℓ2pGn`1q ďL2 n`1 Kn`1 ÿ i“1 ››››› Kn ÿ j“1 gn`1 ij pTn`1qPn`1pf n j q ´ Kn ÿ j“1 gn`1 ij pTn`1qPn`1phn j q ››››› 2 ℓ2pGn`1q “L2 n`1 Kn`1 ÿ i“1 ››››› Kn ÿ j“1 gn`1 ij pTn`1q “ Pn`1pf n j q ´ Pn`1phn j q ‰ ››››› 2 ℓ2pGn`1q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='We next note ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Kn`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='i“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='››››› ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Kn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='j“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='gn`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ij ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='pTn`1q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Pn`1pf n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='j q ´ Pn`1phn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='j q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‰ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='››››› ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ℓ2pGn`1q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ď ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Kn`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='i“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˜ Kn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='j“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='}gn`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ij ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='pTn`1q}op} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Pn`1pf n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='j q ´ Pn`1phn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='j q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‰ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='}ℓ2pGn`1q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¸2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ď ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˜Kn`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='i“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Kn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='j“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='}gn`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ij ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='pTn`1q}2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='op ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¸ Kn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='j“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='}} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Pn`1pf n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='j q ´ Pn`1phn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='j q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‰ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='}2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ℓ2pGn`1q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ďR2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='n`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˜Kn`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='i“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Kn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='j“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='}gn`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ij ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='pTn`1q}2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='op ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¸ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='}}f n ´ hn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='j }2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='where the second to last step is an application of the Cauchy Schwarz inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Proceeding inductively and using our previously established estimates, this proves the claim for all settings in which Tn is nor normal (using an additional application of the triangle inequality for the case of holomorphic filters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' To prove the claim for normal Tn as well,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' we note that in this setting we have (writing pφα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' λαq|G| α“1 for a normalozed eigenvalue-eigenvector sequence of Tn`1) that we have Kn`1 ÿ i“1 ››››› Kn ÿ j“1 gn`1 ij pTn`1q “ Pn`1pf n j q ´ Pn`1phn j q ‰ ››››› 2 ℓ2pGn`1q “ Kn`1 ÿ i“1 ››››› Kn ÿ j“1 ÿ α gn`1 ij pλαqxφα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' “ Pn`1pf n j q ´ Pn`1phn j q ‰ yℓ2pGn`1qφα ››››› 2 ℓ2pGn`1q “ Kn`1 ÿ i“1 Kn ÿ j“1 ÿ α |gn`1 ij pλαq|2|xφα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' “ Pn`1pf n j q ´ Pn`1phn j q ‰ yℓ2pGn`1q|2 ď ÿ α ˜ÿ i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='j |gijpλαq|2 ¸ Kn ÿ j“1 |xφα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' “ Pn`1pf n j q ´ Pn`1phn j q ‰ yℓ2pGn`1q|2 ď Bn`1Rn`1}}f n ´ hn j }2 Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Here we applied Cauchy Schwarz once more in the second to last step and bounded ˜ÿ i,j |gijpλαq|2 ¸ ď ˜ sup λPσpT q ÿ i,j |gijpλq|2 ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' To see that these bounds are not necessarily tight, we may simply note that if we have a simple one-layer Network as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 9 below, the stability can be tightened to }ΦNpfq ´ ΦNphq}Lout ď LRB ¨ }f ´ h}Lin with with Bn :“ max i“a,bpsupλPσpT q |gipλq|q as opposed to with Bn :“ b supλPσpT q ř i“a,b |gipλq|2 if T is normal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' as an easy calculation shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' F PROOF OF LEMMA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2 We want to prove the following: 17 Figure 9: Sparsely connected Layer Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Let T, rT be operators on on ℓ2pGq , ℓ2p rGq with }T}op, } rT}op ď C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Let J : ℓ2pGq Ñ ℓ2p rGq be arbitrary but linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' With Kg “ ř8 k“1 |ag k|kCk´1 for g entire and Kg “ 1 2π ű BD 1 zγT pzqγ rT pzq|gpzq|d|z| for g holomorphic, we have }gpTqJ ´ Jgp rTq}op ď Kg ¨ }JT ´ rTJ}op Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Let us first verify the claim for entire g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We first note that rT kJ ´ JT k “ rT k´1p rTJ ´ JTq ` p rT k´1J ´ JT k´1qT “ rT k´1p rTJ ´ JTq ` rT k´2p rTJ ´ JTqT ` p rT k´2J ´ JT k´2qT 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Thus, with }T}op, } rT}op ď C we find } rT kJ ´ JT k}op ď kCk´1} rTJ ´ JT}op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' The claim now follows from applying the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Now let us prove the bound for holomorphic g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We first note the following: 1 rT ´ z p rTJ ´ JTq 1 T ´ z “ 1 rT ´ z rTJ 1 T ´ z ´ 1 rT ´ z JT 1 T ´ z “ „ 1 rT ´ z p rT ´ zqJ ` z rT ´ z ȷ 1 T ´ z ´ 1 rT ´ z „ 1 T ´ z pT ´ zqJ ` z T ´ z ȷ “z ˆ J 1 T ´ z ´ 1 rT ´ z J ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Thus we have }gp rTqJ´JgpTq}op ď 1 2π ¿ BD 1 |z|}RzpTq}op}Rzp rTq}op|gpzq|d|z| ď 1 2π ¿ BD 1 |z|γT pzqγ rT pzq|gpzq|d|z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' G PROOF OF THEOREM 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='3 We prove the following generalization of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='3: Theorem G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Let ΦN, rΦN be the maps associated to N-layer graph convolutional networks with the same non-linearities and functional calculus filters, but based on different graph signal spaces ℓ2pGq, ℓ2p rGq, characteristic operators Tn, rTn and connecting operators Pn, rPn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Assume Bn, rBn ď B as well as Rn, rRn ď R and Ln ď L for some B, R, L ą 0 and all n ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Assume that there are identification operators Jn : ℓ2pGnq Ñ ℓ2p rGnq (0 ď n ď N) almost commuting with non- linearities and connecting operators in the sense of } rPnJn´1f ´ JnPnf}ℓ2p r Gnq ď δ2}f}ℓ2pGnq and }ρnpJnfq´Jnρnpfq}ℓ2p r Gnq ď δ1}f}ℓ2pGnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Depending on whether normal or arbitrary characteristic operators are used, define D2 n :“ ř jPKn´1 ř iPKn D2 gn ij or D2 n :“ ř jPKn´1 ř iPKn K2 gn ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Choose D such that Dn ď D for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Finally assume that }JnTn ´ rTnJn}˚ ď δ and with ˚ “ F if both 18 P→l2(Gout) io (Gout) inooperators are normal and ˚ “ op otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Then we have for all f P Lin and with JN the operator that the KN copies of JN induced through concatenation that }rΦpJ0fq ´ JNΦpfq} Ă Lout ď N ¨ rRLDδ ` δ1BR ` δ2BLs ¨ pBRLqN´1 ¨ }f}Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For simplicity in notation, let us denote the hidden representation of J0f in Ă Ln by rf n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='then note the following ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='}Jn`1f n`1 ´ rf n`1} Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Ln`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˝ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Kn`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='i“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='›››››Jn`1ρn`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˜ Kn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='j“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='gn`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ij ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='pTn`1qPn`1pf n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='j q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¸ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='´ ρn`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˜ Kn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='j“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='gn`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ij ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='pTn`1q rPn`1p rf n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='j q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¸››››› ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ℓ2pGn`1q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ď ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˝ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Kn`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='i“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='›››››Jn`1ρn`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˜ Kn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='j“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='gn`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ij ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='pTn`1qPn`1pf n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='j q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¸ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='´ ρn`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˜ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Jn`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Kn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='j“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='gn`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ij ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='pTn`1qPn`1pf n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='j q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¸››››› ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ℓ2pGn`1q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='`L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˝ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Kn`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='i“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='›››››Jn`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Kn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='j“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='gn`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ij ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='pTn`1qPn`1pf n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='j q ´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Kn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='j“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='gn`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ij ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='pTn`1q rPn`1p rf n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='j q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='››››› ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ℓ2pGn`1q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='We can bound the first term by δ1B ¨ R ¨ pBRLqn ¨ }f}Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For the second term we find L ¨ ˝ Kn`1 ÿ i“1 ›››››Jn`1 Kn ÿ j“1 gn`1 ij pTn`1qPn`1pf n j q ´ Kn ÿ j“1 gn`1 ij pTn`1q rPn`1p rf n j q ››››› 2 ℓ2pGn`1q ˛ ‚ 1 2 ďL ¨ ˝ Kn`1 ÿ i“1 ››››› Kn ÿ j“1 pJn`1gn`1 ij pTn`1q ´ gn`1 ij p rTn`1qJn`1qPn`1pf n j q ››››› 2 ℓ2pGn`1q ˛ ‚ 1 2 `LB ˜ Kn ÿ j“1 ›››Jn`1Pn`1pf n j q ´ rPn`1p rf n j q ››› 2 ℓ2pGn`1q ¸ 1 2 Arguing as in the proof of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 we can bound the first term by LD ¨ δR ¨ pBRLqn}f}Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For the second term we find, LB ˜ Kn ÿ j“1 ›››Jn`1Pn`1pf n j q ´ rPn`1p rf n j q ››› 2 ℓ2pGn`1q ¸ 1 2 ď LBδ2pBRLqn ` }Jnf n ´ rf n} Ă Ln arguing as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Iterating from n “ N to n “ 0 then yields the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' H TRANSFERABILITY: GENERAL CONSIDERATIONS We first prove the statement made at the beginning of Section 5 that }pωId ´ ∆δbq´1 ´ pωId ´ ∆δaq´1}op “ Opδa ¨ δbq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' To this end denote the increasing sequence of eigenvalues (counted without multiplicity) of ∆1 by tλiuM i“0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Recall that λ0 “ 0 Denote the sequence of projections on the corresponding eigenspaces by tPiuM i“0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We have for the resolvent that 1 ωId ´ ∆δ “ 1 ωId ´ δ ¨ ∆1 “ M ÿ i“0 1 ω ´ 1 δ λi Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 19 Thus we have for δa, δb small enough that ›››› 1 ωId ´ ∆δa ´ 1 ωId ´ ∆δb ›››› op “ ˇˇˇˇˇ 1 ω ´ 1 δa λ1 ´ 1 ω ´ 1 δb λ1 ˇˇˇˇˇ “ ˇˇˇˇˇλ1 1 δa ´ 1 δb pω ´ 1 δa λ1qpω ´ 1 δb λ1q ˇˇˇˇˇ “ λ1 1 |pω ´ 1 δa λ1qpω ´ 1 δb λ1q| “ Opδa ¨ δbq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Next we note the convergence pωId´∆δq´1 Ñ P0 ¨pω ´0q´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' But this is obvious, since for λi ‰ 0 we have 1 ω ´ λi δ Ñ 0 as δ Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' I PROOFS OF LEMMA 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='3 AND THEOREM 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='4 Lemma I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Let T and rT be characteristic operators on ℓ2pGq and ℓ2p rGq be respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' If these operators are ω-δ-close with identification operator J, and }Rω}op, Rω}op ď C we have }JgpTq ´ gp rTqJ}op ď Kg ¨ }p rRωJ ´ JRωq}op with Kg “ ű BDp1 ` |z ´ ω|γT pzqqp1 ` |z ´ ω|γ rT pzqq|gpzq|d|z| if g is holomorphic and Kg “ }g}F hol ω,C if g P F hol ω,C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' If T and rT are normal as well as doubly ω-δ-close and g P F cont ω,C , we have Kg “ }g}F cont ω,C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We first deal with the statement concerning holomorphic g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' To this end we note that Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='9 of Post (2012) proves } rRzJ ´ JRz}op ď p1 ` |z ´ ω|γT pzqqp1 ` |z ´ ω|γ rT pzqq ¨ } rRωJ ´ JRω}op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' The claim then follows from }JgpTq ´ gp rTqJ}op ď 1 2π ¿ BD |gpzq|} rRzJ ´ JRz}opd|z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For g P F hol ω,C the claim is proved exactly as in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For g P F cont ω,C we note that p rRωqµp rR˚ ωqνJ ´ J pRωqµ pR˚ ωqν “ p rRωqµ ” p rR˚ ωqνJ ´ J pR˚ ωqνı ` rp rRωqµJ ´ JpRωqµs pR˚ ωqν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Together with the result } rT kJ ´ JT k}op ď kCk´1} rTJ ´ JT}op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' established in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2, the claim then follows from the triangle inequality together with the definition of the semi-norm }g}F cont ω,C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' As in the previous section, we state a slightly more general version of our main theorem of this section: Theorem I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Let Φ, rΦ be the maps associated to N-layer graph convolutional networks with the same non-linearities and functional calculus filters, but based on different graph signal spaces ℓ2pGnq, ℓ2p rGnq, characteristic operators Tn, rTn and connecting operators Pn, rPn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Assume Bn, rBn ď B as well as Rn, rRn ď R and Ln ď L for some B, R, L ą 0 and all n ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Assume that there are identification operators Jn : ℓ2pGnq Ñ ℓ2p rGnq (0 ď n ď N) almost commuting with non- linearities and connecting operators in the sense of } rPnJn´1f ´ JnPnf}ℓ2p r Gnq ď δ2}f}ℓ2pGnq and }ρnpJnfq ´ Jnρnpfq}ℓ2p r Gnqδ1}f}ℓ2pGnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' define D2 n :“ ř jPKn´1 ř iPKn K2 gn ij with Kgn ij as in 20 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Choose D such that Dn ď D for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Finally assume that }JnpωId ´ Tnq´1 ´ pωId ´ rTnq´1Jn}op ď δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' If filters in F cont ω,C are used, assume additionally that }JnppωId ´ Tnq´1q˚ ´ ppωId ´ rTnq´1q˚Jn}op ď δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Then we have for all f P Lin and with JN the operator that the KN copies of JN induced through concatenation that }rΦpJ0fq ´ JNΦpfq} Ă Lout ď N ¨ rRLDδ ` δ1BR ` δ2BLs ¨ pBRLqN´1 ¨ }f}Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' The proof proceeds in complete analogy to the one of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' J COLLAPSING STRONG EDGES: PROOFS AND FURTHER DETAILS We utilize the notation introduced in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Beyond this, we denote the positive semi-definite form induced by the energy functional E r G by E r Gpu, vq :“ xu, ∆Gvyℓ2p r Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We further use the notation E r Gpuq :“ E r Gpu, uq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' With E r G “ ÿ αP r GGreek βP r GGreek Ă Wαβ|upαq ´ upβq|2 ` ÿ aP r GLatin bP r GLatin Ă Wab|upaq ´ upbq|2 ` ÿ aP r GLatin βP r GGreek Ă Waβ|upaq ´ upβq|2 ` ÿ αP r GGreek bP r GLatin Ă Wαb|upαq ´ upbq|2 ` ÿ αP r GGreek Ă Wα‹|upαq ´ up‹q|2 ` ÿ βP r GGreek Ă W‹β|up‹q ´ upβq|2 ` ÿ aP r GLatin Ă Wa‹|upaq ´ up‹q|2 ` ÿ bP r GLatin Ă W‹b|up‹q ´ upbq|2 (9) Similar considerations apply when rG is replaced by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Let us next solve the convex optimization program (5) introduced in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='5, restated here for convenience: Definition J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For each g P G, define the signal ψδ g P ℓ2p rGq as the unique solution to the convex optimization program min E r Gpuq subject to uphq “ δhg for all h P rGLatin ď t‹u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' As a first step we note that all entries of ψg are real and non-negative, which follows since each summand in (9) is non-increasing under the map u ÞÑ |u| due to the reverse triangle ||a|´|b|| ď |a´b|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' To find the explicit form of ψg, fix g P rGLatin Ťt‹u and denote by χg P ℓ2p rGq the signal defined by setting it to χηphq “ δhg for h P rGLatin Ťt‹u and ηgpαq “ ηα g with tηα g uαP r GGreek a set of | rGGreek| 21 free parameters in Rď0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We then have E r Gpχgq “2 ÿ aP r GLatin Ă Wag ` 2 ÿ αP r GGreek Ă Wαg|1 ´ ηα g |2 ` 2 ÿ αP r GGreek bP r GLatin Ťt‹u Ă Wαb|ηα g |2 ` ÿ α,βP r GGreek Ă Wαβ|ηα g ´ ηβ g |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' By definition, χg depends smoothly on the parameters tηα g uαP r GGreek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Finding the minimizer of the convex optimization program (5) is then equivalent to finding the values tηα g uαP r GGreek at which we have BE r Gpχgq Bηαg “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We note 1 4 BE r Gpχgq Bηξ g “ ¨ ˚ ˚ ˝Ă Wgξ ` ÿ aP r GLatin a‰g Ťt‹u Ă Wgξ ` ÿ αP r GGreek Ă Wαg ˛ ‹‹‚ηg ξ ´ ÿ αP r GGreek Ă Wαgηg α ´ Ă Wgξ Collecting these equations for all parameters into a matrix equation, we find that the ’Greek entries’ of the vector ψg are given explicitly by ¨ ˚ ˝ ψgpαq ψgpβq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ˛ ‹‚“ ¨ ˚ ˚ ˝ rdα ´Ă Wαβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ´Ă Wβα rdβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ˛ ‹‹‚ ´1 ¨ ¨ ˚ ˝ Ă Wgα Ă Wgβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ˛ ‹‚, (10) with degrees in rG denoted by rdα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Let us denote the restriction of ψδ g to Greek entries, thought of as a vector in C| r GGreek| by ⃗ηδ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Given the degree rdα corresponding to a Greek index, we decompose it as rdα “ rdr α ` Ă Wα‹ ` Vα with rdr α accounting for edges from α to other greek vertices rdr α “ ÿ βP r GGreek Ă Wαβ “ 1 δ ÿ βP r GGreek ωαβ, and Vα accounting for edges from α to Latin vertices Vα “ ÿ aP r GLatin Ă Waα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Recall that we also may write Ă Wα‹ “ 1 δ ωα‹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We may then write ¨ ˚ ˚ ˝ rdα ´Ă Wαβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ´Ă Wβα rdβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ˛ ‹‹‚“ ¨ ˚ ˚ ˝ rdr α ´Ă Wαβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ´Ă Wβα rdr β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ˛ ‹‹‚` 1 δ ¨ ˚ ˚ ˝ ωα‹ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 0 ωβ‹ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ˛ ‹‹‚` ¨ ˚ ˚ ˝ Vα 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 0 Vβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ˛ ‹‹‚ “: 1 δ L ` 1 δ diagp⃗ω‹q ` V, where we made the obvious definitions for the matrices L and V and denoted by ⃗ω‹ the vector with entries ωα‹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Let us also use the notation h :“ L ` diagpω‹q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 22 Next we want to establish that h is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For this we first note that that L is the graph Laplacian of the subgraph rGGreek;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' which we assume to be connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Hence L is positive semi-definite with the eigenspace corresponding to the eigenvalue zero being spanned by (entry-wise) constant vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Since all entries of ω‹ are non-negative, the operator h is also positive semi-definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Since we assume that the vertex ‹ is connected to at least one other vertex in rGGreek, there is at least one entry in ⃗ω‹ that is strictly greater than zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We show that this already implies that h is in fact also positive definite and hence invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Indeed, for any ⃗v P C| r GGreek| we have x⃗v, L ¨ ⃗vyC|Ă GGreek| “ x⃗v, h ¨ ⃗vyC|Ă GGreek| ` x⃗v, diagp⃗ω‹q ¨ ⃗vyC|Ă GGreek|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Both terms on the right hand side are non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' If ⃗v is a constant (non-zero) vector, the first term vanishes, but since at least one entry of ω‹ is strictly positive, with all others being non-negative, the second term on the right hand side is strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' If ⃗v is non-constant, the first term on the right hand side is larger than zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Hence h is positive definite and thus invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Similarly one proves that (for any δ ě 0) the operator h ` δV is positive definite and hence invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Thus we now know that the operator 1 δ ph ` δV q “ ¨ ˚ ˚ ˝ rdα ´Ă Wαβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ´Ă Wβα rdβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ˛ ‹‹‚ utilized in (10) is indeed invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We note (again with the restriction of ψδ g to Greek entries thought of as a vector in C| r GGreek| denoted by ⃗ηδ g) that we may equivalently write (10) as ph ` δV q´1⃗ηδ g “ δ ⃗Ă Wg (11) and ⃗Ă Wg :“ ¨ ˚ ˝ Ă Wgα Ă Wgβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ˛ ‹‚ thought of as an element of C| r GGreek|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' To proceed, we now first focus on the case g “ ‹, for which we may write (11) equivalently as ph ` δV q´1⃗ηδ ‹ “ ⃗ω‹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' (12) Since ⃗ω‹ is independent of δ, we may take the limit δ Ñ 0 and arrive at pL ` diagp⃗ω‹qq⃗η0 ‹ “ ⃗ω‹ which is uniquely solved by ⃗η0 ‹ “ p1, 1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='.q ” 1Greek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Since we assume δ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 1, we can now investigate the solution ⃗ηδ g for non-zero δ through perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We write ⃗ηδ ‹ “ 1 r GGreek ´ ⃗ζδ ‹ with ⃗ζ0 ‹ “ 0 and find from (12) – using h ¨ 1Greek “ ⃗ηδ ‹ – the defining equation ⃗ζδ ‹ “ δph ` δV q´1 ¨ V ¨ 1 r GGreek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' From this we obtain the estimate }⃗ζδ ‹}ℓ2p r GGreekq ď }ph ` δV q}op ¨ }V ¨ 1 r GGreek}ℓ2p r GGreekq ¨ δ, where we denote by ℓ2p rGGreekq the space graph signal space C| r GGreek| equipped with node weights trµgugP r GGreek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We note that both h and V are positive semi-definite and we thus obtain λminphq ď λminph ` δV q 23 for the minimal eigenvalues of the respective operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Hence }ph ` δV q´1}op ď }h´1}op, and thus also }⃗ζδ ‹}ℓ2p r GGreekq ď }h´1}op ¨ }V ¨ 1 r GGreek}ℓ2p r GGreekq looooooooooooooooomooooooooooooooooon “:K ¨δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' (13) Since }h´1}op “ 1{λminphq we may write K “ }V ¨ 1Greek}ℓ2p r GGreekq λminphq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' (14) From (11) we know that for g ‰ ‹ we have ⃗ηδ g “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We now also want to bound }⃗ηδ g}ℓ2p r GGreekq in terms of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We will do this by establishing the relationship ÿ gP r GLatin ⃗ηδ g “ ⃗ζδ ‹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' (15) and then utilizing our estimate on }⃗ζδ ‹}ℓ2p r GGreekq established above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' To prove (15), we will need the concept of harmonic extensions: Definition J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Denote by ℓ2p rGLatin Y t‹uq the graph signal space C| r GLatinYt‹u| equipped with the node weights trµgugP r GLatinYt‹u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Given an arbitrary signal u P ℓ2p rGLatin Y t‹uq a harmonic extension of u to all of ℓ2p rGq is a signal u P ℓ2p rGq satisfying p∆ r Guqpαq “ 0 @α P rGGreek and uphq “ uphq @ h P rGLatin ď t‹u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We first note that the concept of harmonic extensions is both well-defined an well-behaved: Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Fix u P ℓ2p rGLatin Y t‹uq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' There exists a unique harmonic extension u P ℓ2p rGq of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' It is given as the solution to the convex optimization program min E r Gpuq subject to uphq “ δhg for all h P rGLatin ď t‹u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Furthermore if u and v are the harmonic extensions of u and v, then pu ` vq is the (unique) harmonic extension of pu ` vq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We write a signal ψ P ℓ2p rGq as ψ “ pψ, ηq with ψ P ℓ2p rGLatin Y t‹uq and η P ℓ2p rGGreekq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We then notice ψ “ argminE r Gpuq subject to ψphq “ ψphq for all h P rGLatin ď t‹u ôBE r Gpψq Bηα “ 0 @α P rGGreek and ψphq “ ψphq for all h P rGLatin ď t‹u ô ÿ yP r G Ă Wαypψpαq ´ ψpyqq “ 0 @α P rGGreek and ψphq “ ψphq for all h P rGLatin ď t‹u ôp∆ r Gψqpαq “ 0 @α P rGGreek and ψphq “ ψphq for all h P rGLatin ď t‹u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Here, we treated ηα and its complex conjugate as independent variables and used that E r Gp¨q is a real-valued functional for the first equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' As harmonic extensions are thus equivalently characterised as the solutions of convex minimization programs, they are unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' To prove the last statement, we note that by linearity of the graph Laplacian, pu ` vq certainly is a harmonic extension of pu ` vq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Since harmonic extensions are unique, it is the only one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' After this preparatory effort, we are now ready to prove (15): 24 Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For any δ ě 0 the signals t⃗ηδ gugP r GLatin Ťt‹u form a partition of unity of ℓ2p rGGreekq: ÿ gP r GLatin Ťt‹u ⃗ηδ g “ 1 r GGreek (16) Equivalently we have ÿ gP r GLatin ⃗ηδ g “ ⃗ζδ ‹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' As an immediate Corollary we obtain Corollary J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For any δ ě 0 the signals tψδ gugP r GLatin Ťt‹u form a partition of unity of ℓ2p rGq: ÿ gP r GLatin Ťt‹u ⃗ηδ g “ 1 r G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' (17) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Using the ’boundary conditions’ in (5), it is straightforward to verify that (16) is equivalent to (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' From Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='3 we now know that ψδ g, originally characterised as the solution of the problem min E r Gpuq subject to uphq “ δhg for all h P rGLatin ď t‹u, is equivalently characterised as the harmonic extension of uphq “ δhg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' From the last statement of Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='3, we know that ř gP r GLatin Ťt‹u ⃗ηδ g is the unique harmonic extension of ÿ gP r GLatin Ťt‹u δhg “ 1 r GĂ GLatin Ťt‹u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' But this – in turn – is the unique solution of the problem min E r Gpuq subject to uphq “ 1 for all h P rGLatin ď t‹u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Since we have E r Gp1 r Gq “ 0, which is the lowest possible attainable value of E r Gp¨q, and setting u “ 1 r G is compatible with the ’boundary condition’ uphq “ 1 for all h P rGLatin Ťt‹u, we know that is the (unique) harmonic extension of 1 r GLatin Ťt‹u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' By the last statement of Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='3 we thus have ÿ gP r GLatin Ťt‹u ⃗ηδ g “ 1 r G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Having established that we may write ÿ gP r GLatin ⃗ηδ g “ ⃗ζδ ‹, together with the fact that every entry of each ⃗ηδ g is non-negative, we now know that 0 ď ⃗ηδ gpαq, ⃗ζδ ‹ ď 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Furthermore – using our earlier estimate (13) – we now easily obtain ›››››› ÿ gP r GLatin ⃗ηδ g ›››››› ℓ2p r GGreekq ď K ¨ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Hence – by positivity of the entries – we also have for each individual g P rGLatin that ››⃗ηδ g ›› ℓ2p r GGreekq ď K ¨ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 25 For the weights tµδ gugPG we then find rµg ď µδ g ď rµg ` δK ÿ αP r GGreek rµα if g ‰ ‹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We also write rµp rGGreekq :“ ř αP r GGreek rµα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' If g “ ‹, we have rµδ ‹ ` p1 ´ δqrµp rGGreekq ď µδ ‹ ď rµδ ‹ ` rµp rGGreekq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Having set the scene, we are now ready to prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Following Post & Simmer (2017), instead of checking the conditions of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 and Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2 it is instead sufficient to check the following, with J rJ as defined in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2 to establish Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='6: Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' In addition to identification operators J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' rJ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' assume that there exist additional operators J1 : ℓ2pGq Ñ ℓ2p rGq and rJ1 : ℓ2p rGq Ñ ℓ2pGq so that the following set of equations is satisfied with ϵ “ Opδ 1 2 q }Jf} ď p1 ` ϵ1q}f},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' |xJf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' uy ´ xf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' rJuy| ď ϵ1}f} (18) }f ´ rJJf} ď ϵ1a }f}2 ` EGpfq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' }u ´ J rJu} ď ϵ1b }u}2 ` E r Gpuq (19) }J1f ´ Jf} ď ϵ1a }f}2 ` EGpfq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' } rJu ´ rJ1u} ď ϵ1b }u}2 ` E r Gpuq (20) }E r GpJ1f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' uq ´ EGpf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' rJ1uq} ď ϵ1 ¨ a }f}2 ` EGpfq ¨ b }u}2 ` E r Gpuq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' (21) Then the (normal) operators ∆ and r∆ are (doubly) (-1)- (ϵ “ 12ϵ1) -close with identification-operator J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Here, we always have u P ℓ2p rGq and f P ℓ2pGq) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' This follows immediately after combining Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='12 with Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='15 of Post (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We set J1f “ Jf and p rJ1uqpxq “ upxq and now determine the individual ϵ “ ϵpδq values for which these equations are satisfied: Left-hand-side of (18): For the left hand side of (18) we note (using 2ab ď a2 ` b2 and the fact that the ψg form a partition of unity): }Jf}2 ℓ2p r Gq “ ÿ h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='gPG xψδ h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ψδ gyℓ2p r Gqfphqfpgq ď 1 2 ÿ hPG |fphq|2 ÿ gPG xψδ h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ψgyℓ2p r Gq ` 1 2 ÿ gPG |fpgq|2 ÿ hPG xψδ h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ψδ gyℓ2p r Gq “ 1 2 ÿ hPG |fphq|2xψδ h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 1yℓ2p r Gq ` 1 2 ÿ gPG |fpgq|2x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ψδ gyℓ2p r Gq “ ÿ gPG |fpgq|2µδ g “ }f}2 ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Here the second to last inequality follows from the definition of the weights µδ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Thus the left hand side of (18) holds with ϵ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 26 Right-hand-side of (18): The right hand side of (18) holds trivially with ϵ “ 0 since we have chosen J˚ “ rJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Left-hand-side of (19): Now let us check the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' of (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We have: pf ´ rJJfqpyq “ fpyq ´ ÿ gPG fpgq xψδ g, ψδ yyℓ2p r Gq µδy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Using the constant K defined in (14) we have rµg ď µδ g ď rµg ` δK ÿ αP r GGreek rµα if g ‰ ‹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We also write rµp rGGreekq :“ ř αP r GGreek rµα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' If G “ ‹, we have rµ‹ ` p1 ´ δqrµp rGGreekq ď µδ ‹ ď rµ‹ ` 1rµp rGGreekq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We next note xψδ x, ψδ yyℓ2p r Gq “ rµxδxy ` x⃗ηδ x, ⃗ηδ yyℓ2pGGreekq with Ă Wx the vector with entries Ă Wxpgq “ Ă Wxg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Thus for y ‰ ‹ we find |pf ´ rJJfqpyq| ď ˆ 1 ´ rµy µδy ˙ |fpyq| ` ˇˇˇˇˇˇˇ ÿ gPG g‰y fpgq xψδ g, ψδ yyℓ2p r Gq µδy ˇˇˇˇˇˇˇ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We thus find }f ´ rJJf}ℓ2pGq ď g f f f f e ÿ yPG y‰‹ ¨ ˚ ˝ ˆ 1 ´ rµy µδy ˙ |fpyq| ` ˇˇˇˇˇˇˇ ÿ gPG g‰y fpgq xψδg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ψδyyℓ2p r Gq µδy ˇˇˇˇˇˇˇ ˛ ‹‚ 2 ` ˇˇˇˇˇfp‹q ´ ÿ gPG fpgq xψδ g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ψδ ‹yℓ2p r Gq µδ‹ ˇˇˇˇˇ ď g f f f e ÿ yPG y‰‹ ˆˆ 1 ´ rµy µδy ˙ |fpyq| ˙2 ` g f f f f e ÿ yPG y‰‹ ¨ ˚ ˝ ˇˇˇˇˇˇˇ ÿ gPG g‰y fpgq xψδg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ψδyyℓ2p r Gq µδy ˇˇˇˇˇˇˇ ˛ ‹‚ 2 ` ˇˇˇˇˇfp‹q ´ ÿ gPG fpgq xψδ g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ψδ ‹yℓ2p r Gq µδ‹ ˇˇˇˇˇ To bound the first term of the estimate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' we note (for y ‰ ‹) and δ small enough: ˆ 1 ´ rµy µδy ˙ ď ˜ 1 ´ rµy rµy ` δKrµp rGGreekq ¸ “ δKrµp rGGreekq δKrµy ` rµp rGGreekq ď δ Krµp rGGreekq min gP r GLatin rµg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 27 We also note (for y ‰ ‹) |fpyq| ď 1 min gP r GLatin ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='µg |fpyq|?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='µy ď 1 min gP r GLatin a rµy |fpyq|?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='µy Thus we find g f f f e ÿ yPG y‰‹ ˆˆ 1 ´ rµy µδy ˙ |fpyq| ˙2 ď δ ¨ ˚ ˚ ˝ Krµp rGGreekq min gP r GLatin rµ 3 2g ˛ ‹‹‚ g f f e ÿ yPG y‰‹ |fpyq|2µy ď δ ¨ ˚ ˚ ˝ Krµp rGGreekq min gP r GLatin rµ 3 2g ˛ ‹‹‚}f}ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' To estimate the second term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' we estimate |fpgq| ď 1 min gP r GLatinYt‹u a rµg }f}ℓ2pGq to obtain ˇˇˇˇˇˇˇ ÿ gPG g‰y fpgq xψδ g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ψδ yyℓ2p r Gq µδy ˇˇˇˇˇˇˇ ď ¨ ˚ ˝ 1 min gP r GLatinYt‹u a rµg ˛ ‹‚}fpyq}ℓ2pGq ¨ ˇˇˇˇˇˇˇ ÿ gPG g‰y xψδ g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ψδ yyℓ2p r Gq µδy ˇˇˇˇˇˇˇ “ ¨ ˚ ˝ 1 min gPGLatinYt‹u a rµg ˛ ‹‚}fpyq}ℓ2pGq ¨ ˇˇˇˇˇˇˇ ÿ gPG g‰y x⃗ηδ g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ⃗ηδ yyℓ2p r GGreekq µδy ˇˇˇˇˇˇˇ ď ¨ ˚ ˝ 1 min gP r GLatinYt‹u a rµg ˛ ‹‚}fpyq}ℓ2pGq ¨ ˇˇˇˇˇˇˇ ÿ gPG g‰y x⃗ηδ g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ⃗ηδ yyℓ2p r GGreekq rµy ˇˇˇˇˇˇˇ Thus we find (using that x⃗ηδ g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ⃗ηδ yyℓ2p r GGreekq is a non-negative number and we have } ¨ }2 ď } ¨ }1) g f f f f e ÿ yPG y‰‹ ¨ ˚ ˝ ˇˇˇˇˇˇˇ ÿ gPG g‰y fpgq xψδg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ψδyyℓ2p r Gq µδy ˇˇˇˇˇˇˇ ˛ ‹‚ 2 ď 1 min gP r GLatinYt‹u a rµg }f}ℓ2pGq ¨ ÿ yPG y‰‹ ÿ gPG g‰y x⃗ηδ g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ⃗ηδ yyℓ2p r GGreekq rµy ď 1 min gP r GLatinYt‹u rµ 3 2g }f}ℓ2pGq ¨ ÿ yPG y‰‹ ÿ gPG g‰y x⃗ηδ g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ⃗ηδ yyℓ2p r GGreekq ď 1 min gP r GLatinYt‹u rµ 3 2g }f}ℓ2pGq ¨ ÿ yPG y‰‹ ÿ gPG x⃗ηδ g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ⃗ηδ yyℓ2p r GGreekq ď 1 min gP r GLatinYt‹u rµ 3 2g }f}ℓ2pGq ¨ x1 r GGreek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ⃗ζδ ‹yℓ2p r GGreekq ď 1 min gP r GLatinYt‹u rµ 3 2g }f}ℓ2pGq ¨ }1 r GGreek}ℓ2p r GGreekq ¨ }⃗ζδ ‹}ℓ2p r GGreekq ď δ ¨ ¨ ˚ ˚ ˝ K ¨ b rµp rGGreekq min gP r GLatinYt‹u rµ 3 2g ˛ ‹‹‚}f}ℓ2pGq Let us thus turn to the remaining term;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' corresponding to y “ ‹: We have 28 ˇˇˇˇˇfp‹q ´ ÿ gPG fpgq xψδ g, ψδ ‹yℓ2p r Gq µδ‹ ˇˇˇˇˇ ď ˇˇˇˇˇ1 ´ xψδ ‹, ψδ ‹yℓ2p r Gq µδ‹ ˇˇˇˇˇ |fp‹q| ` ˇˇˇˇˇˇˇ ÿ gPG g‰‹ fpgq xψδ g, ψδ ‹yℓ2p r Gq µδ‹ ˇˇˇˇˇˇˇ (22) We first deal with the left summand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We note ˇˇˇˇˇ1 ´ xψδ ‹,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ψδ ‹yℓ2p r Gq µδ‹ ˇˇˇˇˇ “ ˇˇˇˇˇ µδ ‹ ´ rµ‹ ´ x1 r GGreek ´ ⃗ζδ ‹,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 1 r GGreek ´ ⃗ζδ ‹yℓ2p r GGreekq µδ‹ ˇˇˇˇˇ ď ˇˇˇˇˇ µδ ‹ ´ rµ‹ ´ x1 r GGreek ´ ⃗ζδ ‹,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 1 r GGreek ´ ⃗ζδ ‹yℓ2p r GGreekq rµ‹ ` rµp rGGreekq ´ δKrµp rGGreekq ˇˇˇˇˇ ď ˇˇˇˇˇˇ ´ µδ ‹ ´ rµ‹ ´ x1 r GGreek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 1 r GGreekyℓ2p r GGreekq ¯ ` ´ x⃗ζδ ‹,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ⃗ζδ ‹yℓ2p r GGreekq ´ 2x1 r GGreek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ⃗ζδ ‹yℓ2p r GGreekq ¯ rµ‹ ` rµp rGGreekq ´ δKrµp rGGreekq ˇˇˇˇˇˇ ď pδKq ` ˇˇˇx⃗ζδ ‹,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ⃗ζδ ‹yℓ2p r GGreekq ´ 2x1 r GGreek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ⃗ζδ ‹yℓ2p r GGreekq ˇˇˇ rµ‹ ` rµp rGGreekq ´ δKrµp rGGreekq ď pδKq ` δ2K2 ` 2}1 r GGreek}ℓ2p r GGreekq ¨ }⃗ζδ ‹}ℓ2p r GGreekq rµ‹ ` rµp rGGreekq ´ δKrµp rGGreekq ď pδKq ` ˇˇˇx⃗ζδ ‹,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ⃗ζδ ‹yℓ2p r GGreekq ´ 2x1 r GGreek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ⃗ζδ ‹yℓ2p r GGreekq ˇˇˇ rµ‹ ` rµp rGGreekq ´ δKrµp rGGreekq ď pδKq ` δ2K2 ` 2 b rµp rGGreekqKδ rµ‹ ` rµp rGGreekq ´ δKrµp rGGreekq ď pδKq ` δ2K2 ` 2 b rµp rGGreekqKδ rµ‹ Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' under the assumption δ ď 1 (implying δ2 ď δ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' we have ˇˇˇˇˇ1 ´ xψδ ‹,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ψδ ‹yℓ2p r Gq µδ‹ ˇˇˇˇˇ ď K ` K2 ` 2 b rµp rGGreekqK rµ‹ ¨ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' This implies that we have ˇˇˇˇˇfp‹q ´ ÿ gPG fpgq xψδ g, ψδ ‹yℓ2p r Gq µδ‹ ˇˇˇˇˇ ď δ ¨ K ` K2 ` 2 b rµp rGGreekqK rµ 3 2‹ ¨ }f}ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 29 For the right-hand-side summand of the estimate in (22) we note ˇˇˇˇˇˇˇ ÿ gPG g‰‹ fpgq xψδ g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ψδ ‹yℓ2p r Gq µδ‹ ˇˇˇˇˇˇˇ “ ˇˇˇˇˇˇˇ ÿ gPG g‰‹ fpgq x⃗ηδ g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ⃗ηδ ‹yℓ2p r GGreekq µδ‹ ˇˇˇˇˇˇˇ ď 1 min gP r GLatinYt‹u rµ 3 2g }f}ℓ2pGq ÿ gPG g‰‹ x⃗ηδ g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ⃗ηδ ‹yℓ2p r GGreekq ď 1 min gP r GLatinYt‹u rµ 3 2g }f}ℓ2pGq ÿ gPG x⃗ηδ g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ⃗ηδ ‹yℓ2p r GGreekq “ 1 min gP r GLatinYt‹u rµ 3 2g }f}ℓ2pGqx1 r GGreek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ⃗ζδ ‹yℓ2p r GGreekq δ ¨ ¨ ˚ ˚ ˝ K ¨ b rµp rGGreekq min gP r GLatinYt‹u rµ 3 2g ˛ ‹‹‚}f}ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Putting it all together, we find for δ ď 1 that }f ´ rJJf}ℓ2pGq ď δ ¨ KA ¨ }f}ℓ2pGq with KA :“ ¨ ˚ ˚ ˝ Krµp rGGreekq min gP r GLatin rµ 3 2g ˛ ‹‹‚` 2 ¨ ˚ ˚ ˝ K ¨ b rµp rGGreekq min gP r GLatinYt‹u rµ 3 2g ˛ ‹‹‚` K ` K2 ` 2 b rµp rGGreekqK rµ 3 2‹ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Thus the left hand side of (19) holds with ϵ “ KA ¨ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Right-hand-side of (19): Hence let us now check the right hand side of (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We note pu ´ J rJuq “ u ´ ÿ xPG xψδ x, uyℓ2p r Gq µδx ψδ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Let us denote by M the matrix representation M δ “ Id ´ rJJ “ Id ´ ÿ xPG xψδ x, ¨yℓ2p r Gq µδx ψδ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We use the triangle inequality to arrive at ›››pu ´ J rJuq ››› ℓ2p r Gq ď ››M 0 ¨ u ›› ℓ2p r Gq ` ››M δ ´ M 0›› op ¨ }u}ℓ2p r Gq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' (23) Using the fact that for g ‰ ‹ we have ⃗ηδ g Ñ ⃗0 an ⃗η0 ‹ “ 1 r GGreek we find in the (δ Ñ 0)-limit that M 0 “ ˜ 0| r GLatin|ˆ| r GLatin| 0| r GLatin|ˆ| r GGreekYt‹u| 0| r GGreekYt‹u|ˆ| r GLatin| M 0 ¸ with 30 M 0 “ ¨ ˚ ˝ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 1 ˛ ‹‚´ 1 rµp rGGreekq ` rµ‹ ¨ ˚ ˝ rµ‹ rµα rµβ ¨ ¨ ¨ rµ‹ rµα rµβ ¨ ¨ ¨ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ˛ ‹‚ acting on ℓ2p rGGreek Y t‹uq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For any element v P ℓ2p rGq, let us denote its restriction to rGGreek Y t‹uby v P ℓ2p rGGreek Y t‹uq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We thus find ››M 0 ¨ u ››2 ℓ2p r GGreekYt‹u “ xM 0 ¨ u, M 0 ¨ uyℓ2p r GGreekYt‹u “ ÿ iP r GGreekYt‹u ÿ jP r GGreekYt‹u upiqupjq ÿ a,bP r GGreekYt‹u « δia ´ rµi rµp rGGreekq ` rµ‹ ff ¨ rµaδab ¨ « δbj ´ rµj rµp rGGreekq ` rµ‹ ff “ ÿ iP r GGreekYt‹u ÿ jP r GGreekYt‹u upiqupjq ÿ aP r GGreekYt‹u « δia ´ rµi rµp rGGreekq ` rµ‹ ff ¨ « rµaδaj ´ rµarµj rµp rGGreekq ` rµ‹ ff “ ÿ iP r GGreekYt‹u ÿ jP r GGreekYt‹u upiqupjq ˆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ˆ ÿ aP r GGreekYt‹u « rµarµ‹δiaδaj ´ δiarµarµj rµp rGGreekq ` rµ‹ ´ δijrµirµj rµp rGGreekq ` rµ‹ ` rµirµarµj prµp rGGreekq ` rµ‹q2 ff “ ÿ iP r GGreekYt‹u ÿ jP r GGreekYt‹u upiqupjq « rµiδij ´ rµirµj rµp rGGreekq ` rµ‹ ff “ ÿ i,jP r GGreekYt‹u ˜ rµirµj rµp rGGreekq ` rµ‹ ¸ |upiq ´ upjq|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' To proceed, we prove the following Lemma: Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Let i, j P rGGreek Y t‹u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Denote by C r GGreekYt‹upi, jq the minimum number of edges for which ωij ŋ 0 needed to connect i and j by a path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Set C r GGreekYt‹u :“ max i‰jP r GGreekYt‹u C r GGreekYt‹upi, jq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Furthermore set Ω :“ min i‰jP r GGreekYt‹u ωij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We have |upiq ´ upjq| ď δ 1 2 ˜ C r GGreekYt‹u ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Ω ¸ b E r Gpuq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We call C r GGreekYt‹u the connectivity constant of the sub-graph rGGreek Y t‹u and note that it is well-defined since we assume rGGreek Y t‹u to be connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 31 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Fix i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Let ti, g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=', gn, ju be the vertices traversed by a path of minimal length determining C r GGreekYt‹upi, jq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We then have |upiq ´ upjq| ď|upiq ´ upg1q| ` |upg1q ´ upg2q| ` .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ` |upgnq ´ upjq| ďδ 1 2 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Ω ˆb Ă Wig1|upiq ´ upg1q|2 ` b Ă Wg1g2|upg1q ´ upg2q|2 ` .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ` b Ă Wgnj||upgnq ´ upjq|2 ˙ ďδ 1 2 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Ω ´b E r Gpuq ` b E r Gpuq ` .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ` b E r Gpuq ¯ “δ 1 2 C r GGreekYt‹upi, jq ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Ω b E r Gpuq ďδ 1 2 C r GGreekYt‹u ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Ω b E r Gpuq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' With the help of this Lemma we then find ››M 0 ¨ u ›› ℓ2p r Gq ď δ 1 2 C r GGreekYt‹u ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Ω b E r Gpuq ¨ g f f e ÿ i,jP r GGreekYt‹u ˜ rµirµj rµp rGGreekq ` rµ‹ ¸ “ δ 1 2 ¨ ¨ ˝C r GGreekYt‹u ¨ b rµp rGGreekq ` rµ‹ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Ω ˛ ‚¨ b E r Gpuq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' To derive a bound for ››M δ ´ M 0›› op in the second term of the estimate (23), we write M δ ´ M 0 “ ˆ B A A: D ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Here we denote by A: : ℓ2p rGLatinq ÝÑ ℓ2p rGGreek Y t‹uq the adjoint of the operator A : ℓ2p rGGreek Y t‹uq ÝÑ ℓ2p rGLatinq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Clearly }A}op “ }A:|op so that we have ››M δ ´ M 0›› op ď }B}op ` 2 }A}op ` }D}op .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' (24) To bound }B}op we note that B is diagonal and we have B “ ¨ ˚ ˚ ˚ ˝ rµa ´ 1 µδa ´ 1 µ0a ¯ rµb ´ 1 µδ b ´ 1 µ0 b ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ˛ ‹‹‹‚ 32 so that }B}op ď „ max aP r GLatin rµa ˇˇˇˇ 1 µδa ´ 1 µ0a ˇˇˇˇ ȷ “ „ max aP r GLatin rµa ˇˇˇˇ 1 µδa ´ 1 µ0a ˇˇˇˇ ȷ “ „ max aP r GLatin rµa ˇˇˇˇ µδ a ´ µ0 a µδa ¨ µ0a ˇˇˇˇ ȷ ď „ max aP r GLatin rµa ˇˇˇˇ µδ a ´ µ0 a rµ2a ˇˇˇˇ ȷ ď « max aP r GLatin rµa ˇˇˇˇˇ Kδrµp rGGreekq rµ2a ˇˇˇˇˇ ff ď δ ¨ » –K ¨ rµp rGGreekq min aP r GLatin µa fi fl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' To estimate }A}op we note A “ ¨ ˚ ˚ ˚ ˚ ˚ ˝ 0 ⃗ηδ apαq µδ a ⃗ηδ apβq µδ a ¨ ¨ ¨ 0 ⃗ηδ bpαq µδ b ⃗ηδ apβq µδ b ¨ ¨ ¨ 0 ⃗ηδ cpαq µδc ⃗ηδ cpβq µδc ¨ ¨ ¨ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ˛ ‹‹‹‹‹‚ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We can consider the map A : ℓ2p rGGreek Y t‹uq ÝÑ ℓ2p rGLatinq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' as a composition of maps A : ℓ2p rGGreek Y t‹uq Id ÝÑ C| r GGreekYt‹u| A ÝÑ C| r GLatin| Id ÝÑ ℓ2p rGLatinq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For the map Id : ℓ2p rGGreek Y t‹uq Ñ C| r GGreekYt‹u| we find }Id}op “ ˜ min gP r GGreekYt‹u rµg ¸´1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Similarly we find for the map Id : ℓ2p rGLatinq Ñ C| r GLatin| that }Id}op “ ˜ max gP r GLatin rµg ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' To bound the operator norm of the map A : C| r GGreekYt‹u| Ñ C| r GLatin|, we use that the operator-norm is smaller than the maximal column-sum times b | rGGreek Y t‹u|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Hence for A as a map from C| r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹u| to C| r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GLatin| we ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='33 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='find ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='}A}op ď ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='| rGGreek Y t‹u| ¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˝ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='gP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GLatin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='µδg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‹‚¨ max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='» ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='– ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='aP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GLatin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='⃗ηδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='apαq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='fi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='fl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='| rGGreek Y t‹u| ¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˝ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='gP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GLatin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='µδg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‹‚¨ max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='” ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='⃗ζδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‹pαq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ı ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='“ δ ¨ K ¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='| rGGreek Y t‹u| ¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˝ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='gP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GLatin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='µδg ¨ min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rµα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‹‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ď δ ¨ K ¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='| rGGreek Y t‹u| ¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˝ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='gP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GLatin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rµg ¨ max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rµα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‹‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Here we estimated max αP r GGreek ” ⃗ζδ ‹pαq ı ď 1 min αP r GGreek a rµα }⃗ζδ ‹}ℓ2p r GGreekq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' In total, we find for the operator-norm of A : ℓ2p rGGreek Y t‹uq ÝÑ ℓ2p rGLatinq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' that }A}op ď δ ¨ K ¨ b | rGGreek Y t‹u| ¨ ¨ ˚ ˚ ˝ max gP r GLatin rµg min gP r GLatin rµg ¨ max αP r GGreekYt‹u rµ 3 2α ˛ ‹‹‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Thus let us now investigate }D}op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' As before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' let us denote by u P ℓ2p rGGreek Y t‹uq the restriction of an element u P ℓ2p rG to rGGreek Y t‹u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We have }D}op “ ›››››› ÿ xP r GLatinYt‹u xψδ x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ¨yℓ2p r GGreekYt‹uq µδx ψδ x ´ ÿ xP r GLatinYt‹u xψ0 x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ¨yℓ2p r GGreekYt‹uq µ0x ψ0 x ›››››› ď ›››››› ÿ xP r GLatin xψδ x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ¨yℓ2p r GGreekYt‹uq µδx ψδ x ´ ÿ xP r GLatin xψ0 x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ¨yℓ2p r GGreekYt‹uq µ0x ψ0 x ›››››› ` ››››› xψδ ‹,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ¨yℓ2p r GGreekYt‹uq µδ‹ ψδ ‹ ´ xψ0 ‹,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ¨yℓ2p r GGreekYt‹uq µ0‹ ψ0 ‹ ››››› “ ›››››› ÿ xP r GLatin xψδ x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ¨yℓ2p r GGreekYt‹uq µδx ψδ x ›››››› ` ››››› xψδ ‹,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ¨yℓ2p r GGreekYt‹uq µδ‹ ψδ ‹ ´ xψ0 ‹,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ¨yℓ2p r GGreekYt‹uq µ0‹ ψ0 ‹ ››››› .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We note for the matrix representation of the first term, that (with α, β P rGGreek Y t‹u) we have ¨ ˝ ÿ xP r GLatin xψδ x, ¨yℓ2p r GGreekYt‹uq µδx ψδ x ˛ ‚ αβ “ ¨ ˝ ÿ xP r GLatin 1 µδx ⃗ηδ xpαq⃗ηδ xpβqrµβ ˛ ‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 34 Using the ’maximal row sum trick’ complementary to the ’maximal column sum trick’ already used for A above and recalling the definition of the weights µδ g :“ ÿ hP r G ψδ gphq ¨ rµh we find ›››››› ÿ xP r GLatin xψδ x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ¨yℓ2p r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹uq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='µδx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ψδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='›››››› ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ď ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='| rGGreek Y t‹u| ¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='xP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rµx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='xP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rµx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='βP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˝ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˝ ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='xP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GLatin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='µδx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='⃗ηδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='xpαq⃗ηδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='xpβqrµβ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ď ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='| rGGreek Y t‹u| ¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='xP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rµx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='yP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rµy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˝ ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='xP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GLatin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='µδx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='⃗ηδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='xpαq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ď ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='| rGGreek Y t‹u| ¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='xP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rµx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='yP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rµy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˝ ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='xP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GLatin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='⃗ηδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='xpαq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ď ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='| rGGreek Y t‹u| ¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='xP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rµx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='yP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rµy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='⃗ζδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‹pαq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ď ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='| rGGreek Y t‹u| ¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='xP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rµx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='yP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='}⃗ζδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‹|ℓ2p r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ď ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='| rGGreek Y t‹u| ¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='xP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rµx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='yP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ K ¨ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 35 It remains to bound the second term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We find (using ›››ψδ ‹ ››› ℓ2p r GGreekYt‹uq ď ›››ψ0 ‹ ››› ℓ2p r GGreekYt‹uq): ››››› xψδ ‹,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' uyℓ2p r GGreekYt‹uq µδ‹ ψδ ‹ ´ xψ0 ‹,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' uyℓ2p r GGreekYt‹uq µ0‹ ψ0 ‹ ››››› ℓ2p r GGreekYt‹uq ď ›››› ˆ 1 µδ‹ ´ 1 µ0‹ ˙ xψδ ‹,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' uyℓ2p r GGreekYt‹uqψδ ‹ ›››› ℓ2p r GGreekYt‹uq ` 1 µ0‹ ›››xψδ ‹,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' uyℓ2p r GGreekYt‹uqψδ ‹ ´ xψ0 ‹,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' uyℓ2p r GGreekYt‹uqψ0 ‹ ››› ℓ2p r GGreekYt‹uq ď ˇˇˇˇ 1 µδ‹ ´ 1 µ0‹ ˇˇˇˇ ¨ ›››ψδ ‹ ››› 2 ℓ2p r GGreekYt‹uq ¨ }u}ℓ2p r GGreekYt‹uq ` 1 µ0‹ ››› ´ xψδ ‹,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' uyℓ2p r GGreekYt‹uq ´ xψ0 ‹,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' uyℓ2p r GGreekYt‹uq ¯ ψ0 ‹ ` xψδ ‹,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' uyℓ2p r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹uq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ψδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‹ ´ ψ0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‹ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯››› ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ℓ2p r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹uq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ď ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ˇˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='µδ‹ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='´ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='µ0‹ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ˇˇˇˇ ¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='›››ψ0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‹ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='››› ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ℓ2p r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹uq ¨ }u}ℓ2p r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹uq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='`2 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='µ0‹ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='›››ψδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‹ ´ ψ0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‹ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='››› ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ℓ2p r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹uq ¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='›››ψ0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‹ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='››› ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ℓ2p r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹uq ¨ }u}ℓ2p r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹uq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ď ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˝ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='δ ¨ K ¨ rµp rGGreekq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rµ‹ ` rµp rGGreekq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rµ‹ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‚¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rµ‹ ` rµp rGGreekq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ }u}ℓ2p r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹uq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='`2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rµ‹ ` rµp rGGreekq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='δ ¨ K ¨ rµp rGGreekq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rµ‹ ` rµp rGGreekq ¨ }u}ℓ2p r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹uq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Thus we find ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='}D}op ď δ ¨ K ¨ rµp rGGreekq ¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˝ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rµ‹ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='` 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rµ‹ ` rµp rGGreekq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='In total,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' using (23) and (24),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' we find ›››pu ´ J rJuq ››› ℓ2p r Gq ďδ 1 2 ¨ ¨ ˝C r GGreekYt‹u ¨ b rµp rGGreekq ` rµ‹ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Ω ˛ ‚¨ b E r Gpuq `δ ¨ » –K ¨ rµp rGGreekq min aP r GLatin µa fi fl ¨ }u}ℓ2p r Gq ` 2 ¨ δ ¨ K ¨ b | rGGreek Y t‹u| ¨ ¨ ˚ ˚ ˝ max gP r GLatin rµg min gP r GLatin rµg ¨ max αP r GGreekYt‹u rµ 3 2α ˛ ‹‹‚¨ }u}ℓ2p r Gq `δ ¨ K ¨ rµp rGGreekq ¨ ¨ ˝ 1 rµ‹ ` 2 1 b rµ‹ ` rµp rGGreekq ˛ ‚¨ }u}ℓ2p r Gq 36 and may hence set ϵ “ δ 1 2 ¨ ¨ ˝C r GGreekYt‹u ¨ b rµp rGGreekq ` rµ‹ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Ω ˛ ‚ `δ ¨ » –K ¨ rµp rGGreekq min aP r GLatin µa fi fl ` 2 ¨ δ ¨ K ¨ b | rGGreek Y t‹u| ¨ ¨ ˚ ˚ ˝ max gP r GLatin rµg min gP r GLatin rµg ¨ max αP r GGreekYt‹u rµ 3 2α ˛ ‹‹‚ `δ ¨ K ¨ rµp rGGreekq ¨ ¨ ˝ 1 rµ‹ ` 2 1 b rµ‹ ` rµp rGGreekq ˛ ‚ Left-hand-side of (20): The left hand side of (20) is true with ϵ “ 0 by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Right-hand-side of (20): Let us thus check the right hand side of (20): We have p rJu ´ rJ1uqpxq “ 1 µx xψδ x, uyℓ2p r Gq ´ upxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We note }p rJu ´ rJ1uq}ℓ2pGq ď ˇˇˇˇ 1 µδ‹ xu, ψδ ‹y ´ up‹q ˇˇˇˇ b µδ‹ ` g f f f e ÿ xPG g‰‹ ˇˇˇˇ 1 µx xψδx, uyℓ2p r Gq ´ upxq ˇˇˇˇ 2 µδx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' (25) We first deal with the left hand term of the estimate and note that for x “ ˚ we have µδ ‹ ď µ0 ‹ “ rµ‹ ` rµp rGGreekq and in the limit δ Ñ 0 that ˇˇˇˇ 1 µδ‹ xψδ ‹,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' uyℓ2p r Gq ´ up‹q ˇˇˇˇ ÝÑ 1 rµ‹ ` rµp rGGreekq ˇˇˇˇˇˇ » – ÿ gP r GGreekYt‹u upgq fi fl ´ up‹q ˇˇˇˇˇˇ “ 1 rµ‹ ` rµp rGGreekq ˇˇˇˇˇˇ ÿ gP r GGreekYt‹u upgq ´ up‹q ˇˇˇˇˇˇ ď 1 rµ‹ ` rµp rGGreekq ÿ gP r GGreekYt‹u |upgq ´ up‹q| ď 1 rµ‹ ` rµp rGGreekq ÿ gP r GGreekYt‹u δ 1 2 ˜ C r GGreekYt‹u ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Ω ¸ b E r Gpuq ďδ 1 2 ¨ | rGGreek Y t‹u| rµ‹ ` rµp rGGreekq ˜ C r GGreekYt‹u ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Ω ¸ b E r Gpuq 37 Here we applied Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Comparing the δ ą 0 and δ “ 0 terms, we find ˇˇˇˇ 1 µδ‹ xψδ ‹, uyℓ2p r Gq ´ 1 µ0‹xψ0 ‹, uyℓ2p r Gq ˇˇˇˇ ď 1 µδ‹ ˇˇˇxψδ ‹ ´ ψ0 ‹, uyℓ2p r Gq ˇˇˇ ` ˇˇˇˇ 1 µδ‹ ´ 1 µ0‹ ˇˇˇˇ ¨ ˇˇˇxψ0 ‹, uyℓ2p r Gq ˇˇˇ ď 1 rµ‹ }u}ℓ2p r Gq ¨ }⃗ζδ ‹}ℓ2p r GGreekq ` ˇˇˇˇ 1 µδ‹ ´ 1 µ0‹ ˇˇˇˇ ¨ ´ rµ‹ ` rµp rGGreekq ¯ }u}ℓ2p r G ďKδ rµ‹ }u}ℓ2p r Gq ` ¨ ˝ Kδ rµ‹ ´ rµ‹ ` rµp rGGreekq ¯ ˛ ‚¨ ´ rµ‹ ` rµp rGGreekq ¯ }u}ℓ2p r G “δ 2K rµ‹ }u}ℓ2p r Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Thus we have ˇˇˇˇ 1 µδ‹ xu, ψδ ‹y ´ up‹q ˇˇˇˇ b µδ‹ ďδ 1 2 ¨ | rGGreek Y t‹u| b rµ‹ ` rµp rGGreekq ˜ C r GGreekYt‹u ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Ω ¸ b E r Gpuq `δ 2K rµ‹ }u}ℓ2p r Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For the remaining term in (25) we note g f f e ÿ xP r GLatin ˇˇˇˇ 1 µx xψδx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' uyℓ2p r Gq ´ upxq ˇˇˇˇ 2 ď g f f e ÿ xP r GLatin ˇˇˇˇ1 ´ rµx µδx ˇˇˇˇ 2 ¨ |upxq|2µδx ` ÿ xP r GLatin ˇˇˇxψδ x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' uyℓ2p r GGreekYt‹uq ˇˇˇ b µδx ďKδ rµ‹ ¨ }u}ℓ2p r Gq ` ÿ xP r GLatin xψδ x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' |u|yℓ2p r GGreekYt‹uq b µδx ďKδ rµ‹ ¨ }u}ℓ2p r Gq ` }⃗ζδ ‹}ℓ2p r GGreekq ¨ „ max xP r GLatin b µδx ȷ }u}ℓ2p r Gq ďKδ rµ‹ ¨ }u}ℓ2p r Gq ` δKrµp rGGreekq ¨ „ max xP r GLatin b Ă µx ` δKrµp rGGreekq ȷ }u}ℓ2p r Gq ďKδ rµ‹ ¨ }u}ℓ2p r Gq ` δKrµp rGGreekq ¨ «c max xP r GLatin Ă µx ` b δKrµp rGGreekq ff }u}ℓ2p r Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Equation (21): It finally only remains to prove the energy differences of (21) and establish |E r GpJ1f, uq ´ EGpf, rJ1uq| ď ϵ ¨ a }f}2 ` EGpfq ¨ b }u}2 ` E r Gpuq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 38 We note that the (unique) operator associated to the energy EG via EGpg, fq “ xg, ∆Gfyℓ2pGq is given by p∆Gfqpxq “ 1 µx ÿ y„Gx Wxypfpxq ´ fpyqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Here the notation "y „G x" signifies that nodes x and y are connected within G through edges with positive edge-weights Wxy ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Similarly the operator associated to E r G via E r Gpv, uq “ xv, ∆ r Guyℓ2p r Gq is given by p∆ r Guqpxq “ 1 rµx ÿ y„Ă Gx Ă Wxypupxq ´ upyqq with the equivalence relation „ r G precisely signifying that Ă Wxy ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' As before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' let us denote by u P ℓ2p rGGreekYt‹uq the restriction of an element u P ℓ2p rG to rGGreekYt‹u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We note EGpψx, uq “ xψx, ∆Guyℓ2pGq “ ÿ y„Gx Wxypupxq ´ upyqq on the smaller graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For the graph rG we find E r Gpψx, uq “ ÿ y„Ă Gx Ă Wxypupxq ´ upyqq ` ÿ αP r GGreek ⃗ηδ xpαq ÿ y„Ă Gα Ă Wαypupαq ´ upyqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Remembering that we have J1f “ Jf “ ÿ xPG fpxqψx and p rJ1uqpxq “ upxq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' we note ˇˇˇE r GpJ1f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' uq ´ EGpf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' rJ1uq ˇˇˇ ď ˇˇˇˇˇˇ ÿ xP r GLatinYt‹u fpxq “ E r Gpψx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' uq ´ EGpψx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' uq ‰ ˇˇˇˇˇˇ ď ¨ ˚ ˚ ˝ 1 c min xP r GLatinYt‹u rµx ˛ ‹‹‚¨ }f}ℓ2pGq ¨ ÿ xP r GLatinYt‹u ˇˇE r Gpψx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' uq ´ EGpψx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' uq ˇˇ Let us first bound the terms corresponding to x ‰ ‹: We have EGpψx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' uq “ ÿ y„Gx y‰‹ Wxypupxq ´ upyqq ` Wx‹pupxq ´ up‹qq “ ÿ y„Gx y‰‹ Ă Wxypupxq ´ upyqq ` Wx‹pupxq ´ up‹qq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 39 as well as E r Gpψx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' uq “ ÿ y„Ă Gx Ă Wxypupxq ´ upyqq ` ÿ αP r GGreek ⃗ηδ xpαq ÿ y„Ă Gx Wαypupαq ´ upyqq “ ÿ y„Gx y‰‹ Ă Wxypupxq ´ upyqq `Ă Wx‹pupxq ´ up‹qq ` ÿ αP r GGreek Ă Wxαpupxq ´ upαqq ` ÿ αP r GGreek ⃗ηδ xpαq ÿ y„Ă Gx Ă Wαypupαq ´ upyqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Hence (for x ‰ ‹) EGpψx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' uq ´ E r Gpψx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' uq “ Wx‹pupxq ´ up‹qq ´ Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Wx‹pupxq ´ up‹qq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Wxαpupxq ´ upαqq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='⃗ηδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='xpαq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='y„Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Gx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Wαypupαq ´ upyqq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˝ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Wxα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‚pupxq ´ up‹qq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Wxαpupxq ´ upαqq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='⃗ηδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='xpαq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='y„Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Gα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Wαypupαq ´ upyqq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˝ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Wxαpupαq ´ up‹qq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='loooooooooooooooooomoooooooooooooooooon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='“:Ix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˝ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='⃗ηδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='xpαq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='y„Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Gα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Wαypupαq ´ upyqq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='loooooooooooooooooooooooooomoooooooooooooooooooooooooon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='“:IIx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' (26) For Ix we find – using Lemma J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='7 – that |Ix| ď ¨ ˝ ÿ αP r GGreek Ă Wxα ˛ ‚¨ δ 1 2 ˜ C r GGreekYt‹u ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Ω ¸ b E r Gpuq and hence ÿ xPG x‰‹ |Ix| ď ¨ ˚ ˝ ÿ xPG x‰‹ ÿ αP r GGreek Ă Wxα ˛ ‹‚¨ δ 1 2 ˜ C r GGreekYt‹u ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Ω ¸ b E r Gpuq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 40 To bound |IIx| we note ˇˇˇˇˇˇ ÿ αP r GGreek ⃗ηδ xpαq ÿ y„Ă Gα Ă Wαypupαq ´ upyqq ˇˇˇˇˇˇ “ ˇˇˇˇˇˇ ÿ αP r GGreek ⃗ηδ xpαq ÿ y„Ă Gα b Ă Wαy b Ă Wαypupαq ´ upyqq ˇˇˇˇˇˇ “ ˇˇˇˇˇˇˇ ÿ αP r GGreek ⃗ηδ xpαq » – ÿ y„Ă Gα Ă Wyα fi fl 1 2 ¨ » – ÿ y„Ă Gα Ă Wyα|upαq ´ upyq|2 fi fl 1 2 ˇˇˇˇˇˇˇ ď ÿ αP r GGreek ⃗ηδ xpαq ¨ » – ÿ y„Ă Gα Ă Wyα fi fl 1 2 ¨ b E r Gpuq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Thus we find – using Cauchy-Schwarz – that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='xPG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='x‰‹ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='|IIx| ď ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='xPG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='x‰‹ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='⃗ηδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='xpαq ¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='» ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='– ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='y„Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Gα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Wyα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='fi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='fl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='E r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Gpuq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='⃗ζδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‹pαq ¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='» ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='– ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='y„Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Gα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Wyα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='fi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='fl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='E r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Gpuq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ď ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rµα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ }⃗ζδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‹}ℓ2p r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekq ¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='» ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='y„Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Gα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Wyα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='fi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='fl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='E r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Gpuq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ď ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rµα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ Kδ ¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='» ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='y„Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Gα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Wyα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='fi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='fl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='E r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Gpuq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ď ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rµα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ Kδ ¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rdα ¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='E r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Gpuq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Here we denoted by rdα the degree of the node α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We further note ÿ αP r GGreek rdα “ ÿ αP r GGreek ÿ yP r GLatin Ă Wαy ` 1 δ ÿ αP r GGreek ÿ yP r GGreekYt‹u ωαy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Writing rd1 int :“ ÿ αP r GGreek ÿ yP r GGreekYt‹u ωαy for the sum of ’internal’ degrees of greek nodes within Greek Y t‹u at δ “ 1 and dexternal :“ ÿ αP r GGreek ÿ yP r GLatin Ă Wαy for the ’total connection strength’ between the Greek and Latin sector, we thus find ÿ xPG x‰‹ |IIx| ď r b rd1 int ¨ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' δ ` a dexternal ¨ δs K min αP r GGreek a rµα ¨ b E r Gpuq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' It remains to bound the x “ ‹ term in (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' To this end we note EGpψ‹, uq “ ÿ y„G‹ W‹ypup‹q ´ upyqq 41 and E r Gpψ‹, uq “ ÿ y„Ă G‹ Ă W‹ypup‹q ´ upyqq ` ÿ αP r GGreek ⃗ζδ ‹pαq ÿ y„Ă Gα Ă Wyαpupαq ´ upyqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For the difference of the energy forms we thus find EGpψ‹, uq ´ E r Gpψ‹, uq “ ÿ y„G‹ W‹ypup‹q ´ upyqq ´ ÿ y„Ă G‹ Ă W‹ypup‹q ´ upyqq ´ ÿ αP r GGreek ⃗ηδ ‹pαq ÿ y„Ă Gα Ă Wyαpupαq ´ upyqq “ ÿ y„G‹ W‹ypup‹q ´ upyqq ´ ÿ y„Ă G‹ Ă W‹ypup‹q ´ upyqq ´ ÿ αP r GGreek ⃗ηδ ‹pαq ÿ y„Ă Gα Ă Wyαpupαq ´ upyqq ` ÿ αP r GGreek ÿ y„Ă Gα Ă Wyαpupαq ´ upyqq ´ ÿ αP r GGreek ÿ y„Ă Gα Ă Wyαpupαq ´ upyqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We have ÿ αP r GGreek ÿ y„Ă Gα Ă Wyαpupαq ´ upyqq “ ÿ αP r GGreek Ă W‹αpupαq ´ up‹qq ` ÿ y„Ă Gα yP r GLatin Ă Wyαpupαq ´ upyqq ` ÿ αP r GGreek ÿ y„Ă Gα yP r GGreek Ă Wyαpupαq ´ upyqq loooooooooooooooooooomoooooooooooooooooooon “0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 42 with the last term vanishing by symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' This implies EGpψ‹,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' uq ´ E r Gpψ‹,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' uq “ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='p1 ´ ⃗ηδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‹pαqq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='y„Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Gα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Wyαpupαq ´ upyqq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='` ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='y„G‹ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='yP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GLatin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˝ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Wyα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‚pup‹q ´ upyqq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='y„Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='G‹ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='W‹ypup‹q ´ upyqq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='W‹αpupαq ´ up‹qq ´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='y„Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Gα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='yP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GLatin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Wαypupαq ´ upyqq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='p1 ´ ⃗ηδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‹pαqq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='y„Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Gα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Wyαpupαq ´ upyqq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='` ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='y„Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='G‹ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='yP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GLatin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˝ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Wyα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‚pup‹q ´ upyqq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='y„G‹ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='yP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreek‹ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='W‹ypup‹q ´ upyqq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='W‹αpupαq ´ up‹qq ´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='y„Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Gα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='yP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GLatin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Wαypupαq ´ upyqq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='p1 ´ ⃗ηδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‹pαqq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='y„Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Gα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Wyαpupαq ´ upyqq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='` ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='y„G‹ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='yP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GLatin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˝ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreekYt‹u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Wyα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‚pup‹q ´ upyqq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='αP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GGreek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='y„Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Gα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='yP r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='GLatin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Wαypupαq ´ upyqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Continuing, we find EGpψ‹, uq ´ E r Gpψ‹, uq “ ÿ αP r GGreek p1 ´ ⃗ηδ ‹pαqq ÿ y„Ă Gα Ă Wyαpupαq ´ upyqq ` ÿ αP r GGreek ÿ y„G‹ yP r GLatin Ă Wyαpup‹q ´ upyqq ´ ÿ αP r GGreek ÿ y„Ă Gα yP r GLatin Ă Wαypupαq ´ upyqq “ ÿ αP r GGreek p1 ´ ⃗ηδ ‹pαqq ÿ y„Ă Gα Ă Wyαpupαq ´ upyqq ` ÿ αP r GGreek ÿ y„Ă Gα yP r GLatin Ă Wyαpup‹q ´ upyqq ´ ÿ αP r GGreek ÿ y„Ă Gα yP r GLatin Ă Wyαpupαq ´ upyqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 43 This – in turn – we can write as EGpψ‹, uq ´ E r Gpψ‹, uq “I ` II with I :“ ÿ αP r GGreek ⃗ζδ ‹pαq ÿ y„Ă Gα Ă Wyαpupαq ´ upyqq, and II :“ ÿ αP r GGreek ÿ y„Ă Gα yP r GLatin Ă Wyαpup‹q ´ upαqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For the first term, we find |I| ď }⃗ζδ ‹} min αP r GGreek a rµα ¨ d ÿ αP r GGreek rdα ¨ b E r Gpuq ďr b rd1 int ¨ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' δ ` a dexternal ¨ δs K min αP r GGreek a rµα ¨ b E r Gpuq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For the second term we note |II| ď ÿ αP r GGreek ÿ y„Ă Gα yP r GLatin Ă Wyα|up‹q ´ upαq| ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' δ ¨ ÿ αP r GGreek ÿ y„Ă Gα yP r GLatin Ă Wyα ˜ C r GGreekYt‹u ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Ω ¸ b E r Gpuq “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' δ ¨ dexternal ¨ ˜ C r GGreekYt‹u ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Ω ¸ b E r Gpuq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' K PROOF OF THEOREM 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='7 We prove the following theorem: Theorem K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' In the setting of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='6 denote by T ( rT) adjacency matrices or normalized graph Laplacians on ℓ2pGq (ℓ2pGq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' There are no functions η1, η2 : r0, 1s Ñ Rě0 with ηipδq Ñ 0 as δ Ñ 0 (i “ 1, 2), families of identification operators Jδ, rJδ and ω P C so that Jδ and rJδ are η1pδq-quasi-unitary with respect to rT, T and ω while the operators rT and T remain ω-η2pδq close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We prove these two result through contradiction on a graph with two vertices and one edge with weight 1{δ, which we collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' First fix T ( rT) to be the adjacency matrices Ă W “ ˆ 0 1 δ 1 δ 0 ˙ and W “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' The eigenvectors and eigenvalues of Ă W are given by t´ 1 δ , 1 δ u and v´ “ ˆ 1 ´1 ˙ and v` “ ˆ 1 1 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 44 Denote the orthogonal projections onto the corresponding eigenspaces by tP´, P`u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Take the function g to be defined as gpλq :“ 1 ´ i i ´ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Then since gp0q “ 0 we have gpWq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Furthermore we have gpĂ Wq “ „ 1 ´ i i ´ 1 δ ȷ P` ` „ 1 ´ i i ` 1 δ ȷ P´ “ P` ` P´ ´ δ 1 δ ` iP` ´ δ 1 δ ´ iP´ “ Id ´ δ 1 δ ` iP` ´ δ 1 δ ´ iP´ “ Id „ 1 ´ δ 1 δ ` i ȷ ` „ δ 1 δ ` i ´ δ 1 δ ´ i ȷ P´ “ Id „ 1 ´ δ 1 δ ` i ȷ ´ „ δ 2i δ2 ` 1 ȷ P´ We are interested in ›››gpĂ WqJδ ´ JδgpWq ››› op “ ›››gpĂ WqJδ››› op “ ››››Jδ ´ δ „ 1 δ ` iP` ` 1 δ ´ iP´ ȷ Jδ ›››› op .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Assuming ›››gpĂ WqJδ ´ JδgpWq ››› op “ ›››gpĂ WqJδ››› op ď η1pδq we also find ˇˇˇˇ ››Jδ›› op ˆ i δ ` i ˙ ´ ››JδP´ ›› op ˆ δ2i δ2 ` 1 ˙ˇˇˇˇ ď η1pδq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Thus also ››Jδ›› op ˆ i δ ` i ˙ ď η1pδq ` ››JδP´ ›› op ˆ δ2i δ2 ` 1 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Taking the limit and using the condition }Jδ}op ď 2, we find that ››Jδ›› Ñ 0 as δ Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Since we demand }pJ ´ rJ˚q}op ď η2pδq with lim δÑ0 η2pδq “ 0, we also find } rJ}op “ } rJ˚}op Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Next we note that we have Rω “ 1 ω and demand }pId ´ rJδJδqRω}op Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' However }pId ´ rJδJδqRω}op “ 1 |ω|}Id ´ rJδJδ}op ě 1 |ω|p1 ´ } rJ˚}op}J}opq Ñ 1 |ω| ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 45 Thus we have our contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Hence let us now choose T ( rT) as the normalized graph Laplacians associated to the adja- cency matrices W (Ă W) from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We thus have L “ 0 and Ă L “ ˆ 1 ´1 ´1 1 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' The eigenvectors and eigenvalues of Ă L are given by t0, 2u and v0 “ ˆ 1 1 ˙ and v2 “ ˆ 1 ´1 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Denote the orthogonal projections onto the corresponding eigenspaces by tP0, P2u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Then Ă L “ 2P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Chose a function g such that gp0q “ 0 and without loss of generality assume gp2q “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Then 0 ÐÝ ›››gp Ă L qJδ ´ JδgpL q ››› op “ ››P2Jδ›› op .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' (27) Next we consider the demand }pId ´ Jδ rJδq rRωu} ď η3 ¨ }u}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' (28) Since p Ă L ´ ωIdq is bijective, (28) is implies }pId ´ Jδ rJδqv} ď η3pδq ¨ r|ω|}v} ` } Ă L } ¨ }v}s “ η3pδq ¨ r|ω| ` 2s ¨ }v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' (29) upon writing u “ p Ă L ´ ωIdqv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We also write v “ ˆ va vb ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We write rJδ “ ˆ aδ bδ ˙T and Jδ “ η4pδq ¨ ˆ 1 ´1 ˙ ` fpδq ¨ ˆ 1 1 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' From (27), we know that lim δÑ0 η4pδq “ 0, but we do not yet know the behaviour of fp¨q, aδ, bδ as δ Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' With the above notation, we find from (29) that }pId ´ Jδ rJδqv} “ ›››› ˆ va ´ fpδqaδva ´ fpδqbδvb vb ´ fpδqaδva ´ fpδqbδvb ˙ ´ η4pδq Bˆ va vb ˙ , ˆ aδ bδ ˙F ˆ 1 1 ˙›››› ě ›››› ˆ va ´ fpδqaδva ´ fpδqbδvb vb ´ fpδqaδva ´ fpδqbδvb ˙›››› ´ η4pδq ¨ 4 ¨ }v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 46 Thus, combining this result with (29), we know that ›››› ˆ va ´ fpδqaδva ´ fpδqbδvb vb ´ fpδqaδva ´ fpδqbδvb ˙›››› ÝÑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Thus, since both entries of the above vector need to tend to zero, we need both fpδq ¨ aδ Ñ 1 and fpδq ¨ bδ Ñ 0 as well as fpδq ¨ aδ Ñ 0 and fpδq ¨ bδ Ñ 1 which yields the desired contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' L PROOF OF THEOREM 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='8 We first note how the graph Laplacian ∆GN as we have defined it, is consistent with the underlying positive (in the sense of non-negative eigenvalues) Laplacian ” ´ ∆S1 “ ´ B2 Bθ2 ” on the unit circle S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' To this end, fix 0 ă h ăă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Fix a point x P S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For any suitable function f – by means of Taylor expansions – we may write fpx ` hq “ fpxq ` h ¨ rBθfspxq ` h2 2 ¨ r∆S1fspxq ` Oph3q fpx ´ hq “ fpxq ´ h ¨ rBθfspxq ` h2 2 ¨ r∆S1fspxq ` Oph3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Adding these two terms, we find r´∆S1fspxq “ 2fpxq ´ fpx ` hq ´ fpx ´ hq h2 ` Ophq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' This motivates setting our edgeweights on GN to 1{h2 with h “ 2π{N the distance between evenly spaced nodes on the unit-circle S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Remark L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' It should be noted that this consistency property – while given a heuristic to choose weights – does not (immediately) imply ’convergence’ of ∆GN to ´∆S1 in the sense needed to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' apply Levie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' (2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' As our proof of Theorem L proceeds completely without reference to the limit-circle, we do not proceed beyond the above heuristic in investigating in what (relevant) sense ∆GN approximates ´∆S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We thus now want to prove the following result: Theorem L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' In the large graph setting of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2 choose all node-weights equal to one and N to be odd for definiteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' There exists constants K1, K2 “ Op1q so that for each N ě 1, there exist identification operators J, rJ mapping between ℓ2pGNq and ℓ2pGN`1q so that J and rJ are pK1{Nq-quasi-unitary with respect to ∆GN , ∆GN`1 and ω “ p´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Furthermore, the operators ∆GN and ∆GN`1 are p´1q-pK2{Nq close with identification operator J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We first note that the normalized eigenvectors of GN are given by φN k pxq “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' N ei 2πk N x 0 ď k ă N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' The corresponding eigenvalues are easily found to be λN k “ N 2 π2 sin2 ´ π N ¨ k ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 47 For definiteness, we have assumed N to be odd, so that pN ` 1q is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We define the identification operator J : ℓ2pGNq Ñ ℓ2pGN`1q via JpφN k pxqq “ "φN`1 k for K ă N 2 φN`1 k`1 for K ă N 2 on the orthonormal basis tφN k u0ďkăN and extend it to all of ℓ2pGNq via normality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' This implies that precisely the eigenspace spanned by φN`1 N`1 2 (corresponding to the eigenvalue λN`1 N`1 2 “ pN ` 1q2{π2 ) does not lie in the image of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We set rJ to be the adjoint J˚ of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Choosing ω “ 1, we shall now first check the equations of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Since J is isometric, we have }Jf} “ }f} ď 2}f} as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Since rJ “ J˚, we have } rJ ´ J˚} “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Since rJJ “ Idℓ2pGNq, what remains to be checked is the demand }pId ´ J rJq rR´1}op ď K ¨ 1 N 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We have }pId ´ J rJq rR´1}op “ 1 ¨ 1 1 ` λN`1 N`1 2 “ 1 1 ` N 2{π2 ď π2 pN ` 1q2 ď π2 ¨ 1 N 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Thus let us now check that the conditions of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2 are fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We note that with our identification operator and by symmetry (λN k “ λN N´k), we have }JR´1 ´ rR´1J}op “ max 0ďkă N 2 ˇˇˇˇˇˇ 1 1 ` N 2 π2 sin2 ` π N k ˘ ´ 1 1 ` pN`1q2 π2 sin2 ´ π pN`1qk ¯ ˇˇˇˇˇˇ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We now need to bound the right hand side uniformly in k as N Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' To this end we write a :“ 1{N (which implies N`1 N “ 1 ` a) and x “ k N (which for our allowed values of k implies 0 ď x ă 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='With this we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ˇˇˇˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ` N 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='π2 sin2 ` π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='N k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='˘ ´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ` pN`1q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='sin2 ´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='pN`1qk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ˇˇˇˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='“ pπaq2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ˇˇˇˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='pπaq2 ` sin2 pπxq ´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='pπaq2 ` p1 ` aq2 sin2 ´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='πx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1`a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ˇˇˇˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='“ pπaq2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ˇˇˇˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='p1 ` aq2 sin2 ´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='πx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1`a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='´ sin2 pπxq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rpπaq2 ` sin2 pπxqs ¨ rpπaq2 ` p1 ` aq2 sin2 ´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='πx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1`a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ˇˇˇˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='“ pπaq2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ˇˇˇˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='sin2 ´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='πx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1`a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='´ sin2 pπxq ` a sin2 ´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='πx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1`a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='` a2 sin2 ´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='πx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1`a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rpπaq2 ` sin2 pπxqs ¨ rpπaq2 ` p1 ` aq2 sin2 ´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='πx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1`a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ˇˇˇˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='“ pπaq2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ˇˇˇˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='πx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1`a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='πx a`2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='a`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='` a sin2 ´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='πx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1`a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='` a2 sin2 ´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='πx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1`a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rpπaq2 ` sin2 pπxqs ¨ rpπaq2 ` p1 ` aq2 sin2 ´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='πx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1`a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ˇˇˇˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ď pπaq2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ˇˇˇˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='πx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1`a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='πx a`2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='a`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='` a sin2 ´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='πx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1`a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='` a2 sin2 ´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='πx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1`a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rsin2 ´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='πx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1`a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='s ¨ rpπaq2s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ˇˇˇˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ď a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ˇˇˇˇˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='sinpπx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1`aq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='πx a`2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='a`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='` sin2 ´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='πx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1`a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='` a sin2 ´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='πx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1`a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='sin2 ´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='πx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1`a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ˇˇˇˇˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ď2a ` a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ˇˇˇˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='πx a`2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='a`1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='πx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1`a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ˇˇˇˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ˇˇˇˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='πx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1`a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='a ¨ sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='πx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1`a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ˇˇˇˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Thus we are done if we can show that the function Fpa, xq “ ˇˇˇˇˇˇ sin ´ πx a`2 a`1 ¯ sin ´ πx 1 1`a ¯ ˇˇˇˇˇˇ ¨ ˇˇˇˇˇˇ sin ´ πx a 1`a ¯ a ¨ sin ´ πx 1 1`a ¯ ˇˇˇˇˇˇ is bounded on the rectangle r0, 1s ˆ r0, 1 2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We change variables y “ πx{p1 ` aq and consider Fpa, yq “ ˇˇˇˇ sin pypa ` 2qq sin pyq ˇˇˇˇ ¨ ˇˇˇˇ sin pyaq a ¨ sin pyq ˇˇˇˇ on r0, 1s ˆ r0, π 2 s instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Away from y “ 0 this is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Close to y “ 0 we might Taylor expand in numerators and denominators respectively and then (formally) divide them both respectively by y to see that the function Fpa, yq is indeed regular at y “ 0 too and hence on the entire compact set r0, 1s ˆ r0, π 2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' As a continuous function, F attains its supremum on this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Denote it by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Hence we now know }JR´1 ´ rR´1J}op ď r2 ` Ks ¨ a ” r2 ` Ks ¨ 1 N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Thus we have established the desired Op1{Nq-decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' M PROOF OF THEOREM 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 Theorem M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For p ě 2 we have in the setting of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1 that }Ψp Npfq ´ Ψp Nphq}RKout ď ´śN n“1 LnRnBn ¯ ¨ }f ´ h}Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' In the setting of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='3 or 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='4 and under the additional assumption that the ’final’ identification operator JN satisfies ˇˇ}JNfi}ℓkp r GNq ´ }fi}ℓkpGNq ˇˇ ď 49 δ ¨ K ¨ }fi}ℓ2pGNq for all fi P ℓ2pGNq, we have }Ψp Npfq ´ rΨp NpJ0fq}RKout ď pN ¨ DRL ` K ¨ pBRLqq ¨ pBRLqN´1 ¨ }f}Lin ¨ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' To prove the first claim, we note }Ψp Npfq ´ Ψp Npgq}RKout “ d ÿ iPKout ˇˇ}rΦNpfqsi}ℓppGoutq ´ }rΦNpgqsi}ℓppGoutq ˇˇ2 ď d ÿ iPKout ˇˇ}rΦNpfqsi ´ rΦNpgqsi}ℓppGoutq ˇˇ2 ď d ÿ iPKout ˇˇ}rΦNpfqsi ´ rΦNpgqsi}ℓ2pGoutq ˇˇ2 “ }Φp Npfq ´ Φp Npgq}RKout where we used the reverse triangle inequality and the fact that } ¨ }ℓpp r Goutq ď } ¨ }ℓ2p r Goutq for 2 ď p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' To finish the proof we now only need to apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='To prove the second claim we note ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='}Ψp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Npfq ´ rΨp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='NpJ0fq}RKout ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='d ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='iPKout ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ˇˇˇ}rΦNpfqsi}ℓppGoutq ´ }rrΦNpJ0fqsi}ℓpp r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Goutq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='d ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='iPKout ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ˇˇˇ}rΦNpfqsi}ℓppGoutq ´ }rJNΦNpfqsi}ℓppGoutq ` }rJNΦNpfqsi}ℓppGoutq ´ }rrΦNpJ0fqsi}ℓpp r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Goutq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ď ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='d ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='iPKout ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ˇˇˇ}rΦNpfqsi}ℓppGoutq ´ }JNrrΦNpfqsi}ℓppGoutq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='` ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='d ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='iPKout ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ˇˇˇ}JNrΦNpfqsi}ℓppGoutq ´ }rrΦNpJ0fqsi}ℓpp r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Goutq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ˇˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ď K ¨ δ ¨ }JNΦpfq} Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Lout ` }rΦpJ0fq ´ JNΦpfq} Ă ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Lout ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='and the claim follows as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' The proof of the third claim proceed in complete analogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' N ADDITIONAL DETAILS ON EXPERIMENTAL SETUP Scaling Operators: The adjacency matrix fo the given graph is given by A “ ¨ ˚ ˚ ˚ ˝ 0 16 7 18 19 16 0 6 22 3 7 6 0 1 90 18 22 1 0 23 19 3 90 23 0 ˛ ‹‹‹‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' (30) Collapsing Edges: We consider the setting introduced in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2 and consider a generic fully connected graph rG with | rG| “ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We consider a splitting into rG “ rGLatin Ť rGGreek Ťt‹u with | rGLatin| “ 3 and | rGGreek| “ 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' As described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='2, we assume Ą Wab, Ă Wa‹ “ Op1q, @a, b P rGLatin and Ă Wαβ “ ωαβ δ and Ă Wα‹ “ ωα‹ δ such that pωαβ, ωα‹ “ Op1q for all α, β P rGGreek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' For completeness and reproducibility, the full adjacency matrix Ă W can be found in Appendix N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' We set 50 node weight on rG to one and – as discussed – construct a graph G with |G| “ 4 through ’collapsing strong edges’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' The adjacency matrix of the larger ’un-collapsed’ graph rG we consider in Section 7 is given as follows Ă W “ ¨ ˚ ˚ ˚ ˚ ˚ ˚ ˚ ˚ ˝ 0 4 2 10 4 5 6 7 4 0 17 9 8 9 10 11 2 17 0 42 12 13 14 15 10 9 42 0 16{δ 7{δ 18{δ 19{δ 4 8 12 16{δ 0 6{δ 22{δ 3{δ 5 9 13 7{δ 6{δ 0 1{δ 90{δ 6 10 14 18{δ 22{δ 1{δ 0 23{δ 7 11 15 19{δ 3{δ 90{δ 23{δ 0 ˛ ‹‹‹‹‹‹‹‹‚ (31) The exceptional vertex ‹ here carries index "4" ("‹ “ 4").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Node weights are set to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' The Realm of Large Graphs: We also plot the difference in characteristic operators as opposed to their resolvents: Figure 10: Operator Differences Their distances does not decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Experiments on Molecules: The dataset we consider is the QM7 dataset, introduced in Blum & Reymond (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Rupp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' This dataset contains descriptions of 7165 organic molecules, each with up to seven heavy atoms, with all non-hydrogen atoms being considered heavy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' A molecule is represented by its Coulomb matrix CClmb, whose off-diagonal elements CClmb ij “ ZiZj |Ri ´ Rj| correspond to the Coulomb-repulsion between atoms i and j, while diagonal elements encode a polynomial fit of atomic energies to nuclear charge Rupp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' (2012): CClmb ii “ 1 2Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='4 i For each atom in any given molecular graph, the individual Cartesian coordinates Ri and the atomic charge Zi are also accessible individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' To each molecule an atomization energy - calculated via density functional theory - is associated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' The objective is to predict this quantity, the performance metric is mean absolute error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' Numerically, atomization energies are negative numbers in the range ´600 to ´2200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' The associated unit is rkcal/mols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' 51 106 105 Operator Differences 104 II△Gn+1J- J△Gn llop 103 102 101 100 0 250 500 750 1000 1250 1500 1750 2000 NO NOTATIONAL CONVENTIONS We provide a summary of employed notational conventions: Table 1: Classification Accuracies on Social Network Datasets Symbol Meaning G a graph or a vertex set |G| number of nodes in G µi weight of node i M weight matrix x¨,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' ¨y inner product W adjacency matrix D degree matrix ∆ graph Laplacian L normalized graph Laplacian T generic operator T ˚ adjoint of T σpTq spectrum (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' collection of eigenvalues) of T λ an eigenvalue gpTq function g applied to operator T } ¨ }op operator norm (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' spectral norm) } ¨ }F Frobenius norm ω a complex number ω complex conjugate of ω z a complex number Bϵpωq open ball of radius ϵ around ω ag k,bg k complex number determined by g and indexed by k U open set extending to infinity in C D a Cauchy domain in C BD the boundary of D pωId´Tq´1, Rω the resolvent of T at ω γT p¨q resolvent profile of T ű .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='dz a complex line integral ű .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='d|z| the corresponding real line integral ρ a non-linearity P a connecting operator L (possibly hidden) feature space associated to a GCN Φ map associated to a GCN ϵ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' δ small numbers J an identification operator (possibly dependent on some ϵ or δ) rG Graph consisting of regular nodes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content=' an exceptional node ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='and a strongly connected sub-graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rGGreek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='nodes in a strongly connected sub-graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='‹ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='exceptional node to which a strongly connected sub-graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='is collapsed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='rGLatin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='regular nodes in rG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='EGp¨q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Energy form associated to the (undirected) graph G ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='distance between nodes on the circle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='} ¨ }p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='the p-norm on Rd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='a natural number ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Ψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='graph-level feature map associated to a GCN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Zi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='atomic charge of atom corresponding to node i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='xi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Cartesian position of atom corresponding to node i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='ZiZj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='}xi´xj} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Coulomb interaction between atoms i and j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='}xi ´ xj} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='Euclidean distance between xi and xj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} +page_content='52' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFJT4oBgHgl3EQfFiwP/content/2301.11443v1.pdf'} diff --git a/KdE3T4oBgHgl3EQfAQmY/content/tmp_files/2301.04256v1.pdf.txt b/KdE3T4oBgHgl3EQfAQmY/content/tmp_files/2301.04256v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2539857c0014b9178c4c4edf89e6622be48ed215 --- /dev/null +++ b/KdE3T4oBgHgl3EQfAQmY/content/tmp_files/2301.04256v1.pdf.txt @@ -0,0 +1,792 @@ +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 +∗olaf.baake@gmail.com +†jorge.zanelli@uss.cl +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]. +[10] +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]. +[11] +Roberto Emparan, Antonia Micol Frassino, and Benson Way. “Quantum BTZ black hole”. In: +JHEP 11 (2020), p. 137. doi: 10.1007/JHEP11(2020)137. arXiv: 2007.15999 [hep-th]. +[12] +Matías Briceño, Cristián Martínez, and Jorge Zanelli. “Overspinning naked singularities in +AdS3 spacetime”. In: (May 2021). arXiv: 2105.06488 [gr-qc]. +[13] +Alan R. Steif. “The Quantum stress tensor in the three-dimensional black hole”. In: Phys. Rev. +D 49 (1994), pp. 585–589. doi: 10.1103/PhysRevD.49.R585. arXiv: gr-qc/9308032. +[14] +S. J. Avis, C. J. Isham, and D. Storey. “Quantum Field Theory in anti-De Sitter Space-Time”. +In: Phys. Rev. D 18 (1978), p. 3565. doi: 10.1103/PhysRevD.18.3565. +8 + +[15] +Kiyoshi Shiraishi and Takuya Maki. “Quantum fluctuation of stress tensor and black holes in +three dimensions”. In: Phys. Rev. D 49 (1994), pp. 5286–5294. doi: 10.1103/PhysRevD.49.5286. +arXiv: 1804.07872 [gr-qc]. +[16] +Kiyoshi Shiraishi and Takuya Maki. “Vacuum polarization near asymptotically anti-de Sit- +ter black holes in odd dimensions”. In: Class. Quant. Grav. 11 (1994), pp. 1687–1696. doi: +10.1088/0264-9381/11/7/009. arXiv: 1901.00977 [gr-qc]. +[17] +Yves Decanini and Antoine Folacci. “Off-diagonal coefficients of the Dewitt-Schwinger and +Hadamard representations of the Feynman propagator”. In: Phys. Rev. D 73 (2006), p. 044027. +doi: 10.1103/PhysRevD.73.044027. arXiv: gr-qc/0511115. +9 + diff --git a/KdE3T4oBgHgl3EQfAQmY/content/tmp_files/load_file.txt b/KdE3T4oBgHgl3EQfAQmY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8d94f06fb9ba52f167e017844b83d051e993a694 --- /dev/null +++ b/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'} +page_content=' 1969), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'} +page_content=' 252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'} +page_content=' [2] Máximo Bañados, Claudio Teitelboim, and Jorge Zanelli.' 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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'} +page_content=' 9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE3T4oBgHgl3EQfAQmY/content/2301.04256v1.pdf'} diff --git a/L9E4T4oBgHgl3EQfiw1J/content/tmp_files/2301.05136v1.pdf.txt b/L9E4T4oBgHgl3EQfiw1J/content/tmp_files/2301.05136v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fa0dd6235a98c0c6a97b306f9e0faa336139c8ee --- /dev/null +++ b/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: +magnus@if.ufrgs.br +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 +Aaron, F. D., et al. 2010, JHEP, 05, 032. +Acharya, S., et al. 2020, JHEP, 06, 035. +Acharya, S., et al. 2021, Phys. Lett. B, 820, 136481. +Aid, S., et al. 1996, Nucl. Phys. B, 463, 3–32. +Andreev, V., et al. 2020, Eur. Phys. J. C, 80(12), 1189. +Bauer, T. H., Spital, R. D., Yennie, D. R., & Pipkin, F. 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C, 79(8), 702. + diff --git a/L9E4T4oBgHgl3EQfiw1J/content/tmp_files/load_file.txt b/L9E4T4oBgHgl3EQfiw1J/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3a3321599f48b1fb803a45079fc113678394d527 --- /dev/null +++ b/L9E4T4oBgHgl3EQfiw1J/content/tmp_files/load_file.txt @@ -0,0 +1,468 @@ +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 ∫ 푑2⃗푏 [ 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 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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: csb3752@hunnu.edu.cn +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. +[1] Event +Horizon +Telescope +Collaboration, +K. Akiyama, A. Alberdi, et al., Astrophys. J. +Lett. 875, L1 (2019), arXiv:1906.11238. +[2] Event +Horizon +Telescope +Collaboration, +K. Akiyama, A. Alberdi, et al., Astrophys. 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(2022), 2205.07787. +[205] F. +Ozel, +D. +Psaltis, +and +Z. +Younsi +(2021), +2111.01123. +[206] D. +Psaltis, +Gen. +Rel. +Grav. +51, +137 +(2019), +1806.09740. + diff --git a/LNAyT4oBgHgl3EQfTve8/content/tmp_files/load_file.txt b/LNAyT4oBgHgl3EQfTve8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..438e5b7e40837d4d25a1ea6e349c0367ca98c5b6 --- /dev/null +++ b/LNAyT4oBgHgl3EQfTve8/content/tmp_files/load_file.txt @@ -0,0 +1,2200 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf,len=2199 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='00113v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='HE] 31 Dec 2022 Black hole images: A Review Songbai Chen 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='2 ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Jiliang Jing 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Wei-Liang Qian 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Bin Wang 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='5 1 Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Synergetic Innovation Center for Quantum Effects and Applications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Hunan Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Changsha,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Hunan 410081,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' People’s Republic of China 2 Center for Gravitation and Cosmology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' College of Physical Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Yangzhou University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Yangzhou 225009,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' People’s Republic of China 3 Escola de Engenharia de Lorena,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Universidade de S˜ao Paulo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 12602-810,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Lorena,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' SP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Brazil 4 Faculdade de Engenharia de Guaratinguet´a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Universidade Estadual Paulista,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 12516-410,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Guaratinguet´a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' SP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Brazil 5 School of Aeronautics and Astronautics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Shanghai Jiao Tong University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Shanghai 200240,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' People’s Republic of China In recent years,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' unprecedented progress has been achieved regarding black holes’ observation through the electromagnetic channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' In this review, we briefly summarize the recent developments on the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The potential applications of the relevant studies are also addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Key words: black hole shadow, accretion disk, polarization image PACS numbers: 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='Cs, 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='Mw, 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='Lf I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' It provided direct visual evidence of black hole in our universe, which means that black hole is no longer just a theo- retical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' One of the most important ingredients in the im- ages is the black hole shadow [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The light sources in many theoretical investigations are assumed to ho- mogeneously distribute in the total celestial sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' However, in the real astronomical environment, ac- ∗Corresponding author: csb3752@hunnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='cn cretion disk is a kind of actual and feasible light sources around black holes due to its electromagnetic emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Undoubtedly, the matter configuration and the accretion process in the disks are indelibly im- printed on black hole images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The literature related to black hole images is rapidly increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Therefore, it is necessary to sum- marize the existing works at the present time and make prospects for the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' This review focuses on the black hole images, and their rapid develop- ment and potential applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Next, we introduce the images of the black holes surrounded by a thin disk and their polarization pat- terns arising from synchrotron radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' We also discuss the aspects of the potential applications of black hole images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 2 observer photon sphere rsh FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' However, for the objects with strong gravity, such as the black hole, the situation is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' According to general relativity, light rays travelling in a black hole spacetime can be deflected due to the gravitational field of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' This phenomenon is known as gravitational lensing, which is analogous to optical lensing [17–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The deflection angle of the light ray increases with the decreasing of its impact parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Therefore, it is easy to infer that the light rays passing very close to the black hole will be captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Black hole shadow is a dark silhouette observed in the sky originating from these captured light rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Actually, for a Schwarzschild black hole, its shadow is about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='5 times as large as the event horizon in angular size [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' This can be explained by two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Firstly, it is not only the photons near the event horizon can be captured by a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The light rays entered the photon sphere will be captured by the black hole as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Thus, the bound- ary of the shadow is determined by the photon sphere rather than the event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Features of black hole shadows The shape and size of black hole shadows depend on the black hole parameters and the observer incli- nation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' For a Schwarzschild black hole, the shadow is a perfect disk for the observer with arbitrary incli- nation [10, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' For a rotating Kerr black hole, the shadow also presents a circular silhouette for the ob- server located on its rotation axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' However, for the FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 2: The eyebrowlike shadows near the primary shadow for a Kerr black hole with scalar hair [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' observer in the equatorial plane, the shadow gradu- ally becomes a “D”-shaped silhouette with increasing black hole spin [10, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' This is also true for black holes with Proca hair [15, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Thus, the dependence of shadows on black hole parameters could provide a potential tool to identify black holes in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' It also triggers the further study of black hole shadows in various theo- ries of gravity [25–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Black hole shadow also depends on the (non- )integrability of motion equation of photon travelling in the background spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=', the energy E, the z-component of the angular momen- tum Lz and the Carter constant Q [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' This en- sures the motion of photons is regular so that black hole shadow has the same shape as the photon sphere surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' In these completely integrable systems, the black hole shadow can be calculated by analytical methods [10, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' In particular, due to chaotic lensing, there are eyebrowlike shadows with the self-similar fractal structure near the primary shadow, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 3 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Formation mechanism of black hole shadows The photon sphere plays an important role in the formation of black hole shadows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Actually, the pho- ton sphere is composed of unstable photon circular orbits around black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The unstable photon cir- cular orbits in the equatorial plane are determined by the effective potential and its derivatives [40, 41], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=', Veff(r) = 0, Veff(r),r = 0 and Veff(r),rr < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' These unstable orbits can also be obtained by a geometric way with Gauss curvature and geodesic curvature in the optical geometry [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' (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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Fundamental photon orbits The unstable photon circular orbits in the equatorial plane are often called light rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Light rings also affect dynamical proper- ties of ultracompact objects [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' For the ultracom- pact objects without horizon [43], light rings often come in pairs, one stable and the other unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The existence of a stable light ring always implies a space- time instability [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' In the Schwarzschild spacetime, light rings are the only bound photon orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Moreover, there also exist the non-planar bound photon orbits with constant r and motion in θ, known as spherical orbits ( see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 2 in [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' These spherical orbits are unstable and completely determine the Kerr black hole shadow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Fundamental photon orbits are the generalization of light rings and spherical orbits in usual stationary and axisymmetric spacetimes [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The definition of fundamental photon orbits is given in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The orbit O is open and it can reach the boundary of the effective potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The orbit C is closed and it can not reach the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The sign +(−) denotes the even (odd) parity of the orbit under the Z2 reflection symmetry around the equa- torial plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' nr is the number of distinct r values at FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 3: Some fundamental photon orbits in the (r, θ)- plane and their classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The grey areas represent forbidden regions for the effective potential [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' where the orbit crosses the equatorial plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' For the light rings lied in the equatorial plane, their nr = 0 because such special kind of orbits never cross the equatorial plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' ns is the number of self-intersection points of the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='3, some fundamental pho- ton orbits and their classification are illustrated in the (r, θ)-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' By making use of the fundamental photon orbits, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Cunha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [15] explained the formation of the cuspy silhouette of a Kerr black hole with Proca hair shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='4(b) shows the spatial trajectories of the ten fundamental photon orbits in Cartesian coordinates, which move around the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Other fundamental photon orbits are non-planar bound photon orbits crossing the equa- torial plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The discontinuity originating from this jump, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=', rperi(C1) > rperi(A4), induces the emergence of the cuspy shadow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' If the shadow related branch is connected, the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='0 Ci3+ 04 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 4: The cuspy shadow (a) and the fundamental pho- ton orbits (b) for the Kerr black hole with Proca hair[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' shadow edge will be smooth with no cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' This feature has been dubbed “ghost shadow” in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' This mechanism is also applied to explain the formation of the cuspy shadow in the Konoplya-Zhidenko rotating non-Kerr black hole spacetime [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' These studies could also be beneficial to test the Kerr hy- pothesis through black hole shadows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Invariant phase space structures The invariant phase space structures are very important for dynam- ical systems because they remain invariant under the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' There are several types of invariant struc- tures including fixed points, periodic orbits and in- variant manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The simplest is the fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' These phase space structures are applied extensively to design space trajectory for various of spacecrafts [52–55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Actually, the light rings on the equatorial plane are the fixed points for the photon motion [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='??' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The stable and unstable invariant manifolds correspond to the cases of Re(µj) < 0 and Re(µj) > 0, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' while the center manifold corresponds to the case with Re(µj) = 0 where the eigenvalues µj are pure imaginary numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Due to the special properties of the invariant manifolds, there is no trajectory cross- ing the invariant manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Points in the unstable (stable) invariant manifold move to the fixed points exponentially in backward (forward) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' These orbits γǫ in the center man- ifold collapse into a fixed point as ǫ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Similarly, the periodic orbits also have their own stable and un- stable manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' These invariant phase space struc- tures are also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 1 in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Obviously, the photon spheres and other periodic orbits can be gen- eralized to the Lyapunov orbits related to the fixed points [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' These concepts in dynamical systems provide a powerful theoretical foundation for understanding the formation of shadows cast by black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Only the Lyapunov family of the spherical orbits near the unstable fixed points are responsible for generating the black hole shadow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' For a Kerr black hole with scalar hair, there are three unstable light rings L1, L2 and L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content="0 CS BJ B3'CJ-C3 A=O SA BS eldsta B3 CA 0 TA VS AA EA 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='0 BJ BS bel!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' B3 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='t 5 A-TA Cs C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='.5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 5: Intersections of the unstable manifolds of L1, L2, and L3 as well as their Lyapunov orbits with the image plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Lyapunov orbits related to L1, L2, and L3 are marked by red, green, and blue dots, respectively [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' spherical [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The numerical simulated shadow in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='5 shows that the complicated and disconnected boundaries of the shadow are completely determined by the Lyapunov spherical orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The electromagnetic radiation emitted by the disk illuminates the background around the black hole and makes the black hole shadow visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Thus, the accretion disk is the actual light source in the formation of black hole shadows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Simulta- neously, due to the strong gravitational lensing near the central black hole, the shape of the accretion disk is heavily distorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Luminet first simulated a photograph of a Schwarzschild black hole with a rotating thin accre- tion disk [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Moreover, one also sees a highly deformed image associated with the bottom of the gaseous disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The number of times of light ray crossing the disk determines the order of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The higher order images are closer to the cen- tral black spot and become thinner and fainter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The inner infinite order image is related to the photon sphere, which represents the actual shadow boundary of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Generally, it is difficult to distin- guish the higher order images optically because they are standing quite closely to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The central black area is the black hole shadow formed by the gravitational lensing and capture of light rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' For a disk around the black hole, the region closer to the horizon is generally brighter because the gas is hotter there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' However, the apparent luminosity of the disk’s image for the distant observer is very different from the intrinsic luminosity in the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' There are two kinds of shift effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The other is the well-known Doppler effect originating from the displacement of the source with respect to the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Doppler effect gives rise to amplification for the approaching source and at- tenuation for the retreating source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Therefore, for a disk rotating counterclockwise around the black hole, the apparent luminosity of the disk in the left side is 000 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content="0-=n U=-s'e6 brighter than that of in the right side [56]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' This strong asymmetry of apparent luminosity is the main signature of the black hole image with a thin accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' In short, the effects from Doppler shift and gravitational redshift drastically modify the luminosity distribution for the observed disk image at large distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The black hole spin hardly affects the shape of the primary image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Moreover, the gravitational field of the accretion disk also affects the propagation of photon and fur- ther modifies the shape of black hole shadow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' A heavy accretion disk yields some new features for the black hole image [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' This is a consequence of the in- creasing light rays’ bending towards the disk due to the increase of disk’s “weight”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' These features also appear in the gravity system of a Schwarzschild black hole surrounded by a massive Bach-Weyl ring [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The chaotic lensing also leads to some distinct differences in the shape of photon sphere and the black hole shadow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Recently, the EHT Collaboration has published the polarimet- ric image of the black hole M87∗ [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The twist- ing polarization patterns revealed the existence of magnetic field near the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Thus, it is very vital to make theoretical analyses and numerical simulations on the polarized images for various black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Thus, in the simulations of the polarized image of a black hole, only the hot disk model is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Thus, the propagations of the polarized light rays depend heavily on the plasma properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' In the low-spin case, there are much more symmetric ring shape patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' In the high-spin case as a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='99MBH, the polarized image of the approach- ing jet disappeared in the low-spin case is clear [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' One can also extract information about circular polarization through analyzing the Stokes quantity V in the black hole images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The circular polarization can be amplified by the Faraday conversion in the well-ordered magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' However, the study of circular polarization images is helpful to understand the polarized information in black hole images more completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' It is shown that the circular polar- ization images are sensitive to the inclination angle [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Moreover, there is a “separatrix” in the circu- lar polarization images and across which the sign of the circular polarization is reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' This can be at- tributed to the helical magnetic field structure in the disk [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' It implies that future full polarization EHT images are quite useful tracers of the magnetic field structures near black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Moreover, with this model, the image of a finite thin disk can be produced by simply summing con- tributions from individual radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The studies [80, 81] also indicates that the ring model image is broadly consistent with the polarization morphology of the EHT image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' This suggests that the significant depolarization from the internal Faraday effects is essential when modeling and in- terpreting the M87∗ image [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Thus, this model has been recently applied to study the polarized images of black holes in various spacetimes [89–93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' In this simple ring model, the calculation of the po- larization vector usually resorts to a so-called Walker- Penrose quantity [94, 95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The signs of Q and U are opposite for two adjacent indirect images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' APPLICATION PROSPECTS OF BLACK HOLE IMAGES The significance of studying black hole images lies in the following aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' In the following, we present some potential application prospects of black hole images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Moreover, the asymmetry in brightness is a ro- bust indicator of the orientation of the spin axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Ac- tually, the magnetic field can generate some features of black hole images [99, 100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' For a rotating black hole immersed in a Melvin magnetic field [99], the shadow becomes oblate for the weak magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' However, in the case with the strong magnetic field, the multiple disconnected shadows emerge, including a middle oblate shadow and many striped shadows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' In fact, the photons moving along the stable pho- ton orbits are trapped and they can’t enter the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The chaotic lensing arising from the magnetic field gives rise to the self-similar fractal structures in the black hole shadows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' These new effects in shadows could provide a new way to probe the magnetic field near black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The images of black holes indicate that the su- permassive black holes in the centers of galaxies are actually surrounded by plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The influence of plasma on the shadows de- pends mainly on the ratio between the plasma fre- quency and the photon frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' If the plasma frequency is smaller than the photon frequency, the shadow is not very much different from the vacuum case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The above analyses are based on an assumption of plasma with radial power-law density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' To fit astronomical FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 6: The observables for the apparent shape of the Kerr black hole Rs and δs = Dcs/Rs [120].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' observations, several observables were constructed by using special points on the shadow boundaries in the celestial coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' For the Kerr black hole, the two observables Rs and δs = Dcs/Rs ( as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='6) are introduced to measure the approximate size of the shadow and its deformation with respect to the refer- ence circle [120], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Recently, the length of the shadow boundary and the local curvature radius are introduced to describe the shadow boundary [121].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The black hole spin and the observer inclination can be constrained by simply measuring the maximum and minimum of the curvature radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Moreover, a topological covariant quantity is analyzed to mea- sure and distinguish different topological structures of the shadows [122, 123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='k, and k is an arbitrarily positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' These distortions are both accurate and robust so they can also be imple- mented to analyse the noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' For a realistic black hole surrounded by an accretion disk, the shadow is imprinted on the image of the accretion flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' How- ever, it is not feasible to predict the details of the brightness profile of the accretion flow image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' This algorithm not only measures the prop- erties of the black hole shadow, but also assesses the statistical significance of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The distinct features of black holes originating from deviation parameters in the alternative theory can help test the general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' It is shown that the shadow becomes prolate for the negative devi- ation parameter and becomes oblate for the posi- tive one [128].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The large deformation parameter in the Konoplya-Zhidenko rotating non-Kerr black hole yields the special cusp-shaped shadow for the equatorial observer [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The similar features of shadows also appear in other non-Einstein theories of gravity including the quadratic degenerate higher- order scalar-tensor theories [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' However, this measurement still ad- mits other possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The 2017 EHT mea- surement for M87∗ places a bound on the deviation parameter −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='6 < α13 < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='9 [132].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='0 < γ1,2 < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='9 [132].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' For the Konoplya-Rezzolla-Zhidenko met- ric [135], the EHT measurements results in the con- straint −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='2 < α1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='3 [132].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' In general, such parametric tests cannot be connected directly to an underlying property of the alternative theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Similarly, it also excludes considerable regions of the space of parameters for the doubly-charged dilaton and the Sen black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The constraints and tests from shadows are complementary to those imposed by observations of gravitational waves from stellar-mass sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Black hole shadow may also provide a way to test binary black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='7 shows the change of the shad- ows with time t during the collision of the two black holes with equal mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' At t = 0, the two black holes are mutually away enough and their shadows are sep- arated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' However, each shadow is a little bit elongated in the α direction because of the interaction between the two black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' At t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='6, the eyebrowlike shad- ows appear around the main shadows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' With the further increase of time, the eye- browlike structures grow and the main shadows ap- proach each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Although not discernible in the figure, in fact there appear the fractal structures of the eyebrows, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=', infinitely many thinner eyebrows at the outer region of these eyebrows as well as at the inner region of the main shadows [147].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The simi- lar eyebrowlike shadows are found in the Majumdar- Papapetrou binary black hole system [149].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 11 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 7: The change of black hole shadows with time t during the collision of two equal mass black holes[147].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Fundamental problems in physics Dark matter The nature of dark matter is one of the most important open fundamental questions of physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Despite extensive observational data supporting its presence on a large scale, dark matter has not been directly detected by any scientific instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Dark matter should influence black hole shadow due to its gravi- tational effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' It is found that the mass of dark matter and its distance over mass distribution lead to larger radius of shad- ows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' However, it is out of the reach of the current astronomical instruments [153].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The space-based VLBI RadioAstron [158] will be able to obtain a resolution of 1 − 10µas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' This is still at least three orders of magnitude lower than the resolution required by the CDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The black hole shadow has been stud- ied for a rotating black hole solution surrounded by superfluid dark matter and baryonic matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Moreover, comparing with the dark matter, the shadow size increases consider- ably with the baryonic mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' This can be understood by the fact that the baryonic matter is mostly located in the galactic center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 1= .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='ar=} 4 4 5 0 5 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content="C=t f=4'é 4 4 5 0 5 {=O [XJ\\H] a." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='1= 412 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 8: The expected axion parameter space probed by polarimetric observations of M87∗ (green) and Sgr A∗ (red) for different position angle precisions [175].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The bounds from CAST [167] (gray) and Supernova 1987A (pale yellow) are shown to make a comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' proposed by using the high spatial resolution and polarimetric measurements of the EHT [175].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='8 presents the axion parameter space which is poten- tially probed by M87∗ and Sgr A∗ for different posi- tion angle precisions [175].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' This method is comple- mentary to the constraints from the black hole spin measurements through gravitational wave detections [176].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Moreover, the possibility of probing ultralight axions by the circular polarization light is also studied in [177].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Extra dimension The possible existence of extra dimensions is one of the most remarkable predictions of the string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='535 Mpc obtained from GW170817 [182].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' More importantly, it is an independent limit from imaging the dark shadow of M87∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='003 from a solar sys- tem test [184].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The angular diam- eter in the EHT observations of M87∗ leads to the constraint on the length of the compact extra di- mension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='03125 mm ≲ l ≲ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='6 mm [189].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Simi- larly, from the observations of Sgr A∗, the constraints 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='28070 mm ≲ l ≲ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='6 mm and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='13115 mm ≲ l ≲ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='6 mm can be given by the upper bounds of the emission ring and the angular shadow diameter re- spectively [189].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' In particular, within these bounds, the rotating 5D black string spacetime is free from the Gregory-Laflamme instability [189].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The real radius of the black shadow is equal to the minimum value of Rsmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=" 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。" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Q○=↓。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' roalc Q0=3。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 0 Vo Tor: S 313 parameter in the non-rotating case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' With the increase of the coupling strength, the umbra of the black hole decreases and the penumbra increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The observed polarized intensity in the bright region is stronger than that in the darker region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The rotational symme- try in polarized patterns gradually vanishes with the increase of the inclination angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Quantum electro- dynamic effects from the Euler-Heisenberg effective Lagrangian on the shadow have been studied in the black hole background [199].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The coupling between a photon and a generic vec- tor field is also introduced to study black hole shadow [200].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' It is found that the black hole shadow in edge-on view also has different appearances for differ- ent frequencies of the observed light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' This is because the coupling form alters the way that the system de- pends on the initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' In particular, such a kind of coupling can affect the motion of photons and phenomenologically depict a violation of equivalence principle [200].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Thus, it is proposed as a mechanism to test the equivalence prin- ciple by analyzing black hole shadows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' These images are composed of black hole shadow and the image of accretion disk around the central black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' The fundamental photon orbits and the invariant phase space structures determine the intrinsic features of the black hole shadow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' However, the visualization of the shadow must resort to the emission in the accretion disk around the black hole in the real astronomical environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' For example, some models show a partially obscured shadow and others present an ap- parently exaggerated shadow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' ACKNOWLEDGMENTS We would like to thank Profs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Carlos Herdeiro and Jieci Wang for their useful comments and sug- gestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' This work was supported by the National Natural Science Foundation of China under Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 12035005, 12275078 and 11875026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [1] Event Horizon Telescope Collaboration, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Akiyama, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Alberdi, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=', Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 875, L1 (2019), arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='11238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [2] Event Horizon Telescope Collaboration, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Akiyama, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Alberdi, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=', Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 875, L2 (2019), arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='11239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [3] Event Horizon Telescope Collaboration, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Akiyama, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Alberdi, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=', Astrophys.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 30, 311 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [19] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Refsdal and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Surdej, Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} 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+page_content=' Herdeiro, Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 50, 42 (2018), 1801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='00860.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [21] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Perlick and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Tsupko, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 947, 1 (2022), 2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='07101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Chen, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Jing, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 74, 097401 (2022), 2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='05855.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Cunha, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Herdeiro, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Radu (2022), 2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='06237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [25] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Amarilla, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Eiroa, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Giribet, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' D 81, 124045 (2010), 1005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='0607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [26] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Dastan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Saffari, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Soroushfar, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Plus 137, 1002 (2022), 1606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='06994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [27] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Long, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Chen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Wang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Jing, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' C 80, 1180 (2020), 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='07508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [28] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Stepanian, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Khlghatyan, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Gurzadyan, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Zenin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 135, 91 (2022), 2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='01115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [30] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Younsi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Psaltis, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' ¨Ozel (2021), 2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='01752.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 15 [31] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Carter, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 174, 1559 (1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [32] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Cunha, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Herdeiro, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Radu, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Runarsson, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 115, 211102 (2015), 1509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='00021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [33] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Vincent, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Gourgoulhon, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Kawashima, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Akiyama, Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Jap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 72, 32 (2020), 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 462, 115 (2016), 1602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='03184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [77] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' K.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Narayan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Sadowski, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Zhu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' McClintock, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Davis, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' McKin- ney, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 414, 1183 (2011), 1102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='0010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [78] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Moscibrodzka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Falcke, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Shiokawa, As- tron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 586, A38 (2016), 1510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='07243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [79] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Nakamura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=', Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 868, 146 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Dexter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Davelaar, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Falcke, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 468, 2214 (2017), 1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='02390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [83] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Eckart, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Schodel, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Meyer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Paumard, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Ott, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Gillessen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Eisenhauer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Martins, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Genzel, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 375, 764 (2007), astro- ph/0611737.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [85] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Zamaninasab et al.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Doeleman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Broderick, 16 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Loeb, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Rogers, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 706, 1353 (2009), 0910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='3893.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [87] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Broderick and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Loeb, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' As- tron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 367, 905 (2006), astro-ph/0509237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [88] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Broderick and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Loeb, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' As- tron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 363, 353 (2005), astro-ph/0506433.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [89] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Qin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Chen, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Jing, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' C 82, 784 (2022), 2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='10138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [90] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Zhu and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Guo (2022), 2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='04777.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [91] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Delijski, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Gyulchev, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Nedkova, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Yazad- jiev (2022), 2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='09455.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [92] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Qin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Zhang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Jing, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 938, 2 (2022), 2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='12034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [93] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Chen, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Jing (2022), 2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='00391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [94] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Walker and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Penrose, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 18, 265 (1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [95] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='08750.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [96] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Bonnor, Zeitschrift fur Physik 190, 444 (1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [97] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Chen, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Jing, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' C 82, 835 (2022), 2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='13696.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [98] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Hou, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Yan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Guo, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Chen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' D 106, 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Wei, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Plus 137, 571 (2022), 2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='14602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [114] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Cunha, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Herdeiro, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Radu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 123, 011101 (2019), 1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='09997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [141] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Abbott, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Abbott, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 116, 241103 (2016), 1606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='04855.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [143] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Abbott, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Abbott, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Abbott, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 118, 221101 (2017), 1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='01812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [144] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Abbott, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Abbott, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Wang, JCAP 07, 015 (2018), 1804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='08110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [154] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Xu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Hou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Gong, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Zhu, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Ha- roon, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' D 100, 044012 (2019), 1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='11803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [157] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [160] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Weinberg, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 40, 223 (1978), URL https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [162] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Di Luzio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Giannotti, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Nardi, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Visinelli, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Pani, Lect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Notes Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 906, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='1 (2015), 1501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='06570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' [171] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Carroll, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Field, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Jackiw, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' D 41, 1231 (1990), URL https://link.' metadata={'source': 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+page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content=' 51, 137 (2019), 1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} +page_content='09740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfTve8/content/2301.00113v1.pdf'} diff --git a/MtE3T4oBgHgl3EQfYwqL/content/tmp_files/2301.04491v1.pdf.txt b/MtE3T4oBgHgl3EQfYwqL/content/tmp_files/2301.04491v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..897cbf8801e358985bceada0c6c251d521d1613d --- /dev/null +++ b/MtE3T4oBgHgl3EQfYwqL/content/tmp_files/2301.04491v1.pdf.txt @@ -0,0 +1,2363 @@ +Managing the Migration to +Post-Quantum-Cryptography +- preprint - +* +Nils von Nethen, Alex Wiesmaier, Oliver Weissmann, a d Nouri Alnahawi +Darmstadt University of Applied Sciences, Germany +Abstract. Cryptographically relevant quantum Computers (CRQC) are +presumably able to break today's prevalent classic cryptographic algo- +rithms. Protocols a d sehendes based on these algorithrns would become +insecure if such CRQCs would become available. Although it is not ex- +actly knoten, whether this will actually happen, organizations ( a d the +IT Society) have to plan on migrating to quantum-resilient cryptographic +measures, also knoten as Post-Quantum Cryptography (PQC). However, +migrating IT Systems a d applications in organizations to Support a d +integrale new Software components is a difficult task. There exists to +the best of nur knowledge no generalized approach to manage such a +complex migration for cryptography used in IT Systems. We present a +process for managing the migration fror classic cryptography to PQC. +Our Solution is based on best practices, challenges, a d problems derived +fror established Software migration approaches. Compared to existing +approaches, nur proposal provides a r e n s to help organizations migrate +to PQC in a manageable manne a d maintain crypto-agility. Thus, nur +process does not only serve as a framework for a one-time adaptation +but also as a blueprint for organizing crypto-agile IT Systems. +Keywords: Post-Quantum Cryptography (PQC) - PQC Migration Man- +agement Process (PMMP) +1 +Introduction +Almosen all IT Systems und applications rely on cryptographic mechanismus to en- +sure their Security against different types of attacks. Asymmetrie cryptographic +scheues, such as RSA a d DH, have been always subject to threats fror ad- +vances in cryptanalysis. These scheues are based on number theoretic hardness +assumptions, which eould be broker, should one und efficient algorithmus for solv- +ing t h e . Sufficiently large quantum Computers are assumed to pose such a Great +threat, especially utilizing the light algorithmus [19]. Therefore, cryptographers +* This research work has been party funded by the German Federal Ministry of Edu- +cation a d Research a d the Hessian State Ministry for Higher Education, Research +a d the Arts weithin their joint Support of the National Research Center for Applied +Cyber-Security ATHENE. +Managing the Migration to +Post-Quantum-Cryptography +- preprint - +* +Nils von Nethen, Alex Wiesmaier, Oliver Weissmann, a d Nouri Alnahawi +Darmstadt University of Applied Sciences, Germany +Abstract. Cryptographically relevant quantum computers (CRQC) are +presumably able to break today's prevalent classic cryptographic algo- +rithms. Protocols and schemes based on these algorithms would become +insecure if such CRQCs would become available. Although it is not ex- +actly known, whether this will actually happen, organizations ( a d the +IT society) have to plan on migrating to quantum-resilient cryptographic +measures, also known as Post-Quantum Cryptography (PQC). However, +migrating IT systems a d applications in organizations to support a d +integrate new software components is a difficult task. There exists to +the best of our knowledge no generalized approach to manage such a +complex migration for cryptography used in IT systems. We present a +process for managing the migration from classic cryptography to PQC. +Our solution is based on best practices, challenges, a d problems derived +from established software migration approaches. Compared to existing +approaches, our proposal provides a means to help organizations migrate +to PQC in a manageable manner a d maintain crypto-agility. Thus, our +process does not only serve as a framework for a one-time adaptation +but also as a blueprint for organizing crypto-agile IT systems. +Keywords: Post-Quantum Cryptography (PQC) - PQC Migration Man- +agement Process (PMMP) +1 +Introduction +Almost all IT systems a d applications rely on cryptographic mechanisms to en- +sure their security against different types of attacks. Asymmetric cryptographic +schemes, such as RSA a d DH, have been always subject to threats from ad- +vances in cryptanalysis. These schemes are based on number theoretic hardness +assumptions, which could be broken, should one find efficient algorithms for solv- +ing them. Sufficiently large quantum computers are assumed to pose such a great +threat, especially utilizing the right algorithms 1191. Therefore, cryptographers +* This research work has been partly funded by the German Federal Ministry of Edu- +cation a d Research a d the Hessian State Ministry for Higher Education, Research +a d the Arts within their joint support of the National Research Center for Applied +Cyber-Security ATHENE. + +2 +von Nethen et al. +Worldwide have been, und are still, developing new eryptographic scheues. This +Can be clearly Seen in the ongoing NIST PQC standardization process [6], the +Goal of which is to establish additional Standard cryptographic scheues so that +they can be Integrated into existing IT Systems. However, adapting and migrat- +ing large Software ínfrastructures to se PQC is an extremely difficult task hat +is accompanied by several requirements a d challenges. +In this paper, we present a process for managing the complex migration +towards PQC in organizations a d IT Systems. Or methodologie observes this +migration similar to any Other (emergency) Software migration process, such as +in the Case of the famous Millennium Bug [22, pp. 80-85]. Based on the related +work presented in Sec. 2, we erst identify the requirements and challenges for +a suceessful migration, as weil as the needs and limitations of organizations. +Building upon hat, a d upon existing migration approaches, we prognose a de- +sign for nur migration management proeess (Sec. 4), which also adopts same of +the weil-studied and established concepts. Moreover, nur management process +a r s at improving the cryptographic agility of an organization. Therefore, we +evaluate nur Solution based on the previously defined eriteria. Thus, we ensure +a process design that fits the needs of the industrie. In the following, nur process +is abbrevíated Voith PMMP (PQC-Migration-Management-Process). +2 +Related work +We present related work grouped i f o (a) recommendations und overview papers +and (b) existing management approaches. The formen Features ideas, challenges, +a d governmental requirements for PQC migration. The latte presents more er +less concrete processes for conducting a migration towards PQC. +2.1 +Recommendations and Overview +NIST In an attempt to "Explore Challenges Associated Voith Adoption a d +Use of Post-Quantum Cryptographic Algorithmus", the NIST Features the plan- +ning of the Migration towards PQC in its White paper on "Getting Ready for +Post-Quantum Cryptography" [2]. However, this planning involves marly "ini- +tial discovery steps for the development of migration roadmaps" [2, pp. 6-7]. In +contrast to the context of this work, the NIST provide a broader View as they +merton that the planning of the migration includes interacting Voith standards- +developing organizations to reise awareness of necessary +hanges. To Start a +migration Inside an organization they suggest prioritizing work by discovering +Systems hat use public-key cryptography [2, lines 187-189] . +Alnahawi et. al The mount of research done on PQC is increasing. Alnahawi +et. al present a Survey that offers an overview on the work done so far. They de- +scribe challenges as weil as already available Solutions for post-quantum-enabled +protocols. As a continuation of their printed publication, they nun a Website1 +1 See https : I/pqc-cma.gitlab.io/cma/ +2 +von Nethen et al. +worldwide have been, and are still, developing new cryptographic schemes. This +can be clearly seen in the ongoing NIST PQC standardization process [61, the +goal of which is to establish additional standard cryptographic schemes so that +they can be integrated into existing IT systems. However, adapting and migrat- +ing large software infrastructures to use PQC is an extremely difficult task that +is accompanied by several requirements and challenges. +In this paper, we present a process for managing the complex migration +towards PQC in organizations and IT systems. Or methodology observes this +migration similar to any other (emergency) software migration process, such as +in the case of the famous Millennium Bug [22, pp. 80-851. Based on the related +work presented in Sec. 2, we first identify the requirements and challenges for +a successful migration, as well as the needs and limitations of organizations. +Building upon that, and upon existing migration approaches, we propose a de- +sign for our migration management process (Sec. ill, which also adopts some of +the well-studied and established concepts. Moreover, our management process +aims at improving the cryptographic agility of an organization. Therefore, we +evaluate our solution based on the previously defined criteria. Thus, we ensure +a process design that fits the needs of the industry. In the following, our process +is abbreviated with PMMP (PQC-Migration-Management-Process). +2 +Related work +We present related work grouped into (al recommendations and overview papers +and (be existing management approaches. The former features ideas, challenges, +and governmental requirements for PQC migration. The latter presents more or +less concrete processes for conducting a migration towards PQC. +2.1 +Recommendations and Overview +NIST In an attempt to "Explore Challenges Associated with Adoption and +Use of Post-Quantum Cryptographic Algorithms", the NIST features the plan- +ning of the migration towards PQC in its white paper on "Getting Ready for +Post-Quantum Cryptography" +121. However, this planning involves mainly "ini- +tial discovery steps for the development of migration roadmaps" [2, pp. 6-7]. In +contrast to the context of this work, the NIST provide a broader view as they +mention that the planning of the migration includes interacting with standards- +developing organizations to raise awareness of necessary changes. To start a +migration inside an organization they suggest prioritizing work by discovering +systems that use public-key cryptography [2, lines 187-1891 . +Alnahawi et. al The amount of research done on PQC is increasing. Alnahawi +et. al present a survey that offers an overview on the work done so far. They de- +scribe challenges as well as already available solutions for post-quantum-enabled +protocols. As a continuation of their printed publication, they run a website1 +1 See https : I/pqc-cma.gitlab.io/cma/ + +PMMP +3 +hat organizes the State of the Art and is intended to possibly refrain up-to-date. +The Website and íts reference were extensívely used +file Writing the paper at +hand. +BSI The German Federal Office for Information Security (BSI) provides recom- +mendations for migrating so PQC in [4]. The Institute suggestiv enabling crypto- +agility to react to changing Security levels of the used cryptography, but it is +not explained how to achieve the needed levels of crypto-agility. However, they +emphasize their position by reeommending the use of hybrid Solutions [4] . +ISARA In [11], the migration to PQC is compared to the migration needed for +the Y2k-bug. The migration is drive by Fisk management, thus involving the +leaders of organizations. They explicitly State that advanees in erypto-analysis +theater classic cryptography [11, pp. 7-l5]. The erst step hey prognose is to +increase the eryptographic agility by increasing cryptographic visibility [11, p. +5]. Additionally, they recommend including cryptographic and quantum crypto- +graphic risks in the cyber Fisk strategy of an organization [11, p. 5]. +Their approach distinguishes between the quantum Fisk [11, pp. l6-23] and +the eryptographie Fisk [11, pp. 7-15]. To manage the quantum Fisk, the organiza- +tion needs to become crypto-agile [11, p. 23], which is quite similar to managing +the eryptographic Fisk as a +hole, but rather focusing on the speeific needs of +PQC. +2.2 +Existing management approaches +Zhang et. al In [23], the lessons learned fror migrating an IBM Db2 database +to sing PQC are discussed. The approach, explained in detail in [22, pp. 85-86], +is inspired fror the migrations mitigating the Y2K-bug [22, p. 84]. +Zhang et. al State that organizations Can Start planning now a d make Sure +hey are prepared, by investing in cryptographic agility. They prognose to se +Software design patterns such as the Factory pattern to e fable replacing crypto- +graphic primitives. Considering the problem of coordínating Voith business part- +ners, the authors refer to the IETF, which is already Working on new Internet +Standards featuring PQC2. They State that the task of migrating to PQC is +a Community task a d that Software practítioners all Over the World have to +handle, espeeially by upgrading the Software they maintain to be crypto-agile. +One of the hardest challenge they eneountered while migrating the IBM Db2 +database was the lack of quality of documentation. Most documents were out +of date so they offen needed to reverse-engineer the application. Additionally, +they offen encountered hard-coded key lengths that needed to be Updated [23, p. +16]. In the end, the migration was sueeessful Voith even slightly better response +times [23, p. 12]. +2 mea https://trac.ietf.org/trac/sec/wiki/PQCAgility +PMMP +3 +that organizes the state of the art and is intended to possibly remain up-to-date. +The website and its references were extensively used while writing the paper at +hand. +BSI The German Federal Office for Information Security (BSI) provides recom- +mendations for migrating to PQC in 141. The institute suggests enabling crypto- +agility to react to changing security levels of the used cryptography, but it is +not explained how to achieve the needed levels of crypto-agility. However, they +emphasize their position by recommending the use of hybrid solutions 141 . +ISARA In [11], the migration to PQC is compared to the migration needed for +the Y2k-bug. The migration is driven by risk management, thus involving the +leaders of organizations. They explicitly state that advances in crypto-analysis +threaten classic cryptography 111, pp. 7-l5]. The first step they propose is to +increase the cryptographic agility by increasing cryptographic visibility 111, p. +5]. Additionally, they recommend including cryptographic and quantum crypto- +graphic risks in the Cyber risk strategy of an organization 111, p. 51. +Their approach distinguishes between the quantum risk [11, pp. 16-23] and +the cryptographic risk 111, pp. 7-15]. To manage the quantum risk, the organiza- +tion needs to become crypto-agile [11, p. 23], which is quite similar to managing +the cryptographic risk as a whole, but rather focusing on the specific needs of +PQC. +2.2 +Existing management approaches +Zhang et. al In [23], the lessons learned from migrating an IBM Db2 database +to using PQC are discussed. The approach, explained in detail in [22, pp. 85-861, +is inspired from the migrations mitigating the YAK-bug [22, p. 841. +Zhang et. al state that organizations can start planning now and make sure +they are prepared, by investing in cryptographic agility. They propose to use +software design patterns such as the factory pattern to enable replacing crypto- +graphic primitives. Considering the problem of coordinating with business part- +ners, the authors refer to the IETF, which is already working on new internet +standards featuring poo? They state that the task of migrating to PQC is +a community task and that software practitioners all over the world have to +handle, especially by upgrading the software they maintain to be crypto-agile. +One of the hardest challenges they encountered while migrating the IBM I)b2 +database was the lack of quality of documentation. Most documents were out +of date so they often needed to reverse-engineer the application. Additionally, +they often encountered hard-coded key lengths that needed to be updated [23, p. +16]. In the end, the migration was successful with even slightly better response +times [23, p. I2]. +2 mea https://trac.ietf.org/trac/sec/wiki/PQCAgility + +4 +von Nethen et al. +la/Iashatan a d Heinzmann In [12], three different pathos are presented for +migrating to PQC. The authors advise to establish a governance model und +body to Follow their recommendations for migrating. Following that, they ad- +vise to assess the risks quantum Computers Wright pose. This is done by exam- +ining the current cryptographic footprint of the organization. Afterwards, the +quantum-resistant +alternatives should be Selected and implemented according +to the Chosen path [12]. +All pathos lead to remediation projects. The authors describe this task as +implementing the newly published eryptography Standards. Organizations that +have Chosen to follow path A will the observe their position in the Geld. Orga- +nizations following path B will the execute their roadmap. And organizations +following path C will the "simply need to make (relatively minor) adjustments +to +hat they already have in place" [12, p. 22]. Additionally, the authors suggest +the organizations will also need to have a deprecation path for the pre-quantum +cryptographic implementations in plaee [12, p. 22]. +3 +Requirements +In This section, the requirements of the developed Migration Management ap- +proach to PQC are described. Defining these requirements is mandatory for the +evaluation afterwards. Problems defined in existing White papers er migration +drafts, as weil as lessons learned fror the migrations presented above [12,23] are +used as a basis for the requirements. +Migration tinıeline As stated in [20, p. 20], the Integration of PQC Can a d +should Start today. In [11, p. 23], it is Generally recommended to Start early, +s i r e cryptographic upgrades are challenging and time-consuming. The Utimaco +GmbH similarly suggestiv putting the topic on the agenda of the organization, in +oder to prevent harvest-then-decrypt +attaeks [21, p. 4]. Furthermore, in [22, p. +16], the authors advise having a Clear roadmap a d a timeline for the migration. +Consequently, the PMMP needs to help deine a timeline for the migration. +It has to be possible for executives to estimate the duration of each step a d +the migration as a +hole. Therefore, each phase of the migration has to have +metrics by which the duration +an be measured, estimated, a d steered. +Security The Migration process needs to ensure +hat a System uses post- +quantum Secure algorithmus afterwards. This m a n s that the algorithmus imple- +mented need to mitigate the Fisk of an attacke +utilizingg a CRQC to decrypt +the organization°s Communication Voith respect to the lifetime of the secret data, +er faking authentication data er digital signatures . +The migration process itself must not allow for new vulnerabilities to open +up +file it is berg executed. For instanz, disabling cryptography modules +completely, because hey Carnot get migrated, must not be an Option a d must +be prevented by the proeess Voith appropriate countermeasures. +Applications +4 +von Nethen et al. +Mashatan and I-Ieinzmann In [121, three different paths are presented for +migrating to PQC. The authors advise to establish a governance model and +body to follow their recommendations for migrating. Following that, they ad- +vise to assess the risks quantum computers might pose. This is done by exam- +ining the current cryptographic footprint of the organization. Afterwards, the +quantum-resistant alternatives should be selected and implemented according +to the chosen path [121. +All paths lead to remediation projects. The authors describe this task as +implementing the newly published cryptography standards. Organizations that +have chosen to follow path A will then observe their position in the field. Orga- +nizations following path B will then execute their roadmap. And organizations +following path C will then "simply need to make (relatively minors adjustments +to what they already have in place" [12, p. 221. Additionally, the authors suggest +the organizations will also need to have a deprecation path for the pre-quantum +cryptographic implementations in place [12, p. 221. +3 +Requirements +In this section, the requirements of the developed migration management ap- +proach to PQC are described. DeNning these requirements is mandatory for the +evaluation afterwards. Problems defined in existing white papers or migration +drafts, as well as lessons learned from the migrations presented above 112,231 are +used as a basis for the requirements. +Migration timeline As stated in [20, p. 20], the integration of PQC can and +should start today. In 111, p. 23], it is generally recommended to start early, +since cryptographic upgrades are challenging and time-eonsuming. The Utimaco +GmbH similarly suggests putting the topic on the agenda of the organization, in +order to prevent harvest-then-decrypt attacks [21, p. ill. Furthermore, in 122, p. +16], the authors advise having a clear roadmap and a timeline for the migration. +Consequently, the PMMP needs to help define a timeline for the migration. +It has to be possible for executives to estimate the duration of each step and +the migration as a whole. Therefore, each phase of the migration has to have +metrics by which the duration can be measured, estimated, and steered. +Security The migration process needs to ensure that a system uses post- +quantum secure algorithms afterwards. This means that the algorithms imple- +mented need to mitigate the risk of an attacker utilizingg a CRQC to decrypt +the organization's communication with respect to the lifetime of the secret data, +or faking authentication data or digital signatures . +The migration process itself must not allow for new vulnerabilities to open +up while it is being executed. For instance, disabling cryptography modules +completely, because they cannot get migrated, must not be an option and must +be prevented by the process with appropriate countermeasures. +Applications + +PMMP +5 +offering both, classic cryptography und PQC must not allow downgrade attacks. +If the Communication Voith a partner is changed to sing PQC, ít must not revert +to legacy algorithmus. Exceptions will be needed, but have to be reviewed a d +accepted. Moreover, the newly implemented post-quantum +Secure algorithmus +have to be used correctly. For example, the process Wright provide processes for +educating participants in the migration (e.g., programmes, administrators, er +project leaders) . +Furthermore, the process needs to handle advancements in cryptographic +analysis, weakening formerly promising PQC algorithmus. In [20, p. 21], it is rec- +ommended to se hybrid methods that +se pre- and post-quantum cryptography +simultaneously. Hybrid methods are also used in the process presented in [11, p. +23], in Order to minimize the Fisk of harvest-and-decrypt +attacks. They prognose +that the process must favor hybrid methods for higher-risk applications a d pro- +mote cryptographie agility in the organization. In [13, p. 41], it is suggested to +"reise hybrids", by which they mea implementing hybrid algorithmus, supporting +post-quantum and classic primitives at the same time, thus taking the best of +both worlds and enabling interoperability between Systems. Therefore, it an be +assumed that hybrid Solutions will probably become the de facto Standard Way +of integrating PQC. +Completeness The Migration process has to ensure that all relevant Systems +that need a migration to PQC actually get migrated. The migration process has +to recommend a mechanisch by which a comprehensive list of Systems needing a +migration +an be compiled [22, p. 85]. +Context awareness In [11, p. 24], it is recommended to ask vendors of cryp- +tographie products about their "quantum-readiness". +For the migration process, +this m a n s that on needs mechanismus that assess the cryptographic agility of +vendors if third-party Systems are used. Most importantly, the migration to PQC +Can only sueceed if Communication partners migrate as weil. The migration pro- +cess is required to ensure hat a context-aware Migration strategy is developed +a d applied. +Additionally, in [23, p. 14], the authors report a steep learning curve for devel- +opers wanting to integrale PQC, i.e. the need to acquire substantiell knowledge +on the topic before starting the actual work. To conquer this Challenge, it is re- +quired to ensure that Software erıgineers involved in the migration are educated +weil enough. In [16] it is pointed out, that employees have to be educated in +the new technology. The migration process has to elaborate on how to educate +workers. For instanz, it is suggested to collaborate Voith universities, as these +provide a large pool of talents [16, p. 199]. This eould be used for the edueation +of the employees a d for helping Voith the migration. +The different post-quantum Secure algorithmus are suitable for different needs. +Approaching a new er in cryptography, there will no langer be on universal +cryptography that Can be used for all se Cases. As required in [20, p. 22], the +migration has to be tailored to the needed Security level of the protected data. +PMMP +5 +offering both, classic cryptography and PQC must not allow downgrade attacks. +If the communication with a partner is changed to using PQC, it must not revert +to legacy algorithms. Exceptions will be needed, but have to he reviewed and +accepted. Moreover, the newly implemented post-quantum secure algorithms +have to be used correctly. For example, the process might provide processes for +educating participants in the migration (e.g., programmers, administrators, or +project leaders) . +Furthermore, the process needs to handle advancements in cryptographic +analysis, weakening formerly promising PQC algorithms. In [20, p. 211, it is rec- +ommended to use hybrid methods that use pre- and post-quantum cryptography +simultaneously. Hybrid methods are also used in the process presented in [11, p. +23], in order to minimize the risk of harvest-and-decrypt +attacks. They propose +that the process must favor hybrid methods for higher-risk applications and pro- +mote cryptographic agility in the organization. In [13, p. 411, it is suggested to +"raise hybrids", by which they mean implementing hybrid algorithms, supporting +post-quantum and classic primitives at the same time, thus taking the best of +both worlds and enabling interoperability between systems. Therefore, it can be +assumed that hybrid solutions will probably become the de facto standard way +of integrating PQC. +Completeness The migration process has to ensure that all relevant systems +that need a migration to PQC actually get migrated. The migration process has +to recommend a mechanism by which a comprehensive list of systems needing a +migration can be compiled 122, p. 85]. +Context awareness In [11, p. 241, it is recommended to ask vendors of cryp- +tographic products about their "quantum-readiness". +For the migration process, +this means that one needs mechanisms that assess the cryptographic agility of +vendors if third-party systems are used. Most importantly, the migration to PQC +can only succeed if communication partners migrate as well. The migration pro- +cess is required to ensure that a context-aware migration strategy is developed +and applied. +Additionally, in [23, p. 141, the authors report a steep learning curve for devel- +opers wanting to integrate PQC, i.e. the need to acquire substantial knowledge +on the topic before starting the actual work. To conquer this challenge, it is re- +quired to ensure that software engineers involved in the migration are educated +well enough. In 1161 it is pointed out, that employees have to be educated in +the new technology. The migration process has to elaborate on how to educate +workers. For instance, it is suggested to collaborate with universities, as these +provide a large pool of talents [16, p. 1991. This could be used for the education +of the employees and for helping with the migration. +The different post-quantum secure algorithms are suitable for different needs. +Approaching a new era in cryptography, there will no longer be one universal +cryptography that can be used for all use cases. As required in [20, p. 22], the +migration has to be tailored to the needed security level of the protected data. + +6 +von Nethen et al. +The decision on which algorithm is used has to be balaneed between Speed, s i e , +und Security. Consequently, the Migration process has to Guide the selection of +suitable algorithmus vor different applications. +Crypto-agility Technical debüts, such as hard-coded key lengths, add further +levels of complexity [22, p. 16]. The BSI recommends implementing crypto- +graphic agility +file new applications are developed er existing ortes are up- +graded. This enables the organization to update the cryptographic primitives +More easily [5, p. 61]. Other organizations, such as ISARA [11, p. 24] er uti- +maco [21, pp. 5-6] also recommend this. Thus, the Migration process must present +mechanismus that an help Voith ensuring a sufficient level of cryptographie agility. +Furthermore, the BSI recommends hybrid Solutions that combine classic +cryptography Voith post-quantum Solutions [5, p. 3]. For high Security areas, +the BSI even requires hybrid Solutions [4, p. 7] +Interoperability a d availability White migrating an application, connected +applications need to stay able to communicate Voith the migrated application. +Interoperabílity needs to be ensured for the organization as a +hole and must not +Interrupt the Operation of the organization°s business processes. As an example, +in [11, p. 9], it is required to komplement "Forward backward compatibility" such +that Systems +an still operator during migration. +Interim resultat Migrating a +hole organization fror sing classic cryptogra- +phy to PQC is a process that Can take quite a +file. The process must be able +to deliver Interim resultat that Can be used before the entre organization is mi- +grated. An example of an Interim result is a Single application that is migrated +to PQC (but is still able to communicate Voith connected applications that have +not yet been migrated). +Rea ct to advancements in cryptanalysis If there are advancements in +cryptanalysis, The process has so provide Solutions that Can be Applied +her +the implemented algorithmus turn out to be insecure. This is very related to the +before-mentioned crypto-agílíty requirement. +4 +Defining the la/Ianagement Process +In This section, we deine nur management process. In the context of this paper +migration r e n s replacing eryptographic primitives Voith those that Can protect +data fror an attacker Voith access to a quantum Computer powerful enough to +break classic cryptography. +To be able to aequire the resources, approval by the decision makers of the +organization is needed. The migration to PQC is a complex task that requires +substantiell resources, e.g. more powerful hardware, time, er external Support. In +6 +von Nethen et al. +The decision on which algorithm is used has to he balanced between speed, size, +and security. Consequently, the migration process has to guide the selection of +suitable algorithms for different applications. +Crypto-agility Technical debts, such as hard-coded key lengths, add further +levels of complexity 122, p. 161. The BSI recommends implementing crypto- +graphic agility while new applications are developed or existing ones are up- +graded. This enables the organization to update the cryptographic primitives +more easily 15, p. 611. Other organizations, such as ISARA 111, p. 241 or uti- +maco 121, pp. 5-61 also recommend this. Thus, the migration process must present +mechanisms that can help with ensuring a suftieient level of cryptographic agility. +Furthermore, the BSI recommends hybrid solutions that combine classic +cryptography with post-quantum solutions 15, p. 31. For high security areas, +the BSI even requires hybrid solutions 14, p. 71 +Interoperability and availability While migrating an application, connected +applications need to stay able to communicate with the migrated application. +Interoperability needs to be ensured for the organization as a whole and must not +interrupt the operation of the organization's business processes. As an example, +in [11, p. 91, it is required to implement "forward backward compatibility" such +that systems can still operate during migration. +Interim results Migrating a whole organization from using classic cryptogra- +phy to PQC is a process that can take quite a while. The process must be able +to deliver interim results that can be used before the entire organization is mi- +grated. An example of an interim result is a single application that is migrated +to PQC (but is still able to communicate with connected applications that have +not yet been migrated). +React to advancements in cryptanalysis If there are advancements in +cryptanalysis, the process has to provide solutions that can be applied when +the implemented algorithms turn out to be insecure. This is very related to the +before-mentioned crypto-agility requirement. +4 +Defining the la/[anagement Process +In this section, we define our management process. In the context of this paper +migration means replacing cryptographic primitives with those that can protect +data from an attacker with access to a quantum computer powerful enough to +break classic cryptography. +To be able to acquire the resources, approval by the decision makers of the +organization is needed. The migration to PQC is a complex task that requires +substantial resources, e.g. more powerful hardware, time, or external support. In + +PMMP +7 +the PQC Migration recommendation by ETSI, in is recommended to create the +rote of a migration manager Who has to manage the migration [8, pp. 15-16]. In +an organization that has an ISMS implemented, this Wright be the ISO, which +is in Charge of managing the organization"s Information Security (See Figure 1). +Organizations that are required to have an ISMS in place a d er are certified +after ISO 27001 Standard, already have the needed Support fror management [7] . +There, Support is required as the management of Information Security happens +fror top to bottom rather than fror bottom to top. To migrate successfully, the +top management has to delegate tasks for the suceessful Integration of PQC. +Figure l g i e s an overview of the PMMP steps and their interdependencíes. +It is highly recommended to execute the steps in their given oder, to assure a +smooth flow. For instanz, Fisk assessment relies on the inventory of cryptography +as an Input. +Organization Management +Application Management +Executive Officer +Risk Officer +Security Officer +Application Developer I +System Administrator +Educaflon +| +| +l +Decide on time +1 +left +1 +1 +Define context of +the organization +› +Educaflon +› Assess resources +› Provide Information +on Systems +› Security policy +- - - - - - - - - - - - - - - - - - - - - - - - - - - | n f o r m a t i o n - - - - - - - - - - - - +› Compile cryptographic +inventory +I +l +Assess risks < +v +Group and +prioritize +applications +information- - - - - - - +› +Fig. 1. PMMP Overview +4.1 +Education +To get Support fror management, it is necessary to educate decision makers +on the topic of post-quantum cryptography to ensure they know why there is a +need to migrate. Education Can take the Form of Workshops er Seminars where +managers have the opportunity to ask questions and understand the dangers. It +is important to ensure the decision makers know what is Coming a d react in a +seeure manne. +PMMP +7 +the PQC migration recommendation by ETSI, it is recommended to create the +role of a migration manager who has to manage the migration [8, pp. 15-161. In +an organization that has an ISMS implemented, this might be the ISO, which +is in charge of managing the organization's information security (see Figure I). +Organizations that are required to have an ISMS in place and or are certified +after IS() 27001 standard, already have the needed support from management [7] . +There, support is required as the management of information security happens +from top to bottom rather than from bottom to top. To migrate successfully, the +top management has to delegate tasks for the successful integration of PQC. +Figure l gives an overview of the PMMP steps and their interdependencies. +It is highly recommended to execute the steps in their given order, to assure a +smooth flow. For instance, risk assessment relies on the inventory of cryptography +as an input. +Organization Management +Application Management +Executive Officer +Risk Officer +Security Officer +Application Developer / +System Administrator +Education +I +I +l +Decide on time +I +left +I +l +Define context of +the organization +> +Education +> Assess resources +> Provide information +on systems +> Security policy +- - - - - - - - - - - - - - - - - - - - - - - - - - - | n f o r m a t i o n - - - - - - - - - - - - +> Compile cryptographic +inventory +I +l +Assess risks < +v +Group and +prioritize +applications +lnformation- - - - - - - +› +Fig. 1. PMMP Overview +4.1 +Education +To get support from management, it is necessary to educate decision makers +on the topic of post-quantum cryptography to ensure they know why there is a +need to migrate. Education can take the form of workshops or seminars where +managers have the opportunity to ask questions and understand the dangers. It +is important to ensure the decision makers know what is coming and react in a +secure manner. + +8 +von Nethen et al. +Additionally, the Software developers responsible for making changes in the +applications hostess by the organization have to be educated. The programmes +have to know how to komplement the needed algorithmus er how to correctly +se +cryptographic libraries. If knowledge of how to develop Secure Software is not +already established, the engineers have to acquire this knowledge. Management +has to provide unding to e fable this type of Security training (See also [9, sec. +4.12]). +In the training is not sufficient er the organization does not have adequatere- +sources for providing that Kind of education, the organization has to acquire +external Support. For instanz, as the topic of PQC is currently under extensive +research, the organization may involve universities er Institutes researching the +topic. +4.2 +Decide on time left +Based on Mosca°s theorem [14], the Senior management estimates the account of +time left before CRQCs are available. This allows the development of a timeline +on which further decisions Can be based. Of course, the decision has to be realistic +and based upon estimates by authorities like the BSI er the NIST. +4.3 +Security poliey a d Goals +The PQC Migration goals should be documented in the organization"s goals a d +need so be published Inside The organization. Additionally, the Security require- +ments for reaching the goals have to be doeumented in an organization-wide +Security policy. Organizations applying an ISMS already have a Security policy +that should be extended aceordingly. +New Software, either developed in-house er provided by third parties, an be +required to Feature a certain level of cryptographic agility which an be used to +(later) komplement PQC. +To Set a Migration timeline, the policy has to deine the date on which CRQCs +are thought to be a real thing. As show +in Figure l, the Security Officer is in +Charge of developing the Security policy based on the management decisions. +The policy an Feature concrete technical Statements Voith which algorithmus the +organization warts to protect which data. For instanz, to exchange Keys Voith a +party processing data requiring a medium level of Security, KYBER-768 Wright +be Chosen. In addition, the Security policy has to document a list of excluded +algorithmus if needed. This has to be coordinated Voith the Communication part- +ners of the organization. Likewise, a list of preferred algorithmus Can be formed. +The decision is up to the organization because they know best which data has +which proteetion requirements. Alternatively, the seeurity poliey can reference a +technical document fror the BSI er the NIST. +4.4 +Context of the organization +The context of the organization deines the scope in which cryptographic prim- +itives need to be upgraded. Also, the scope deines where the organization has +8 +von Nethen et al. +Additionally, the software developers responsible for making changes in the +applications hosted by the organization have to be educated. The programmers +have to know how to implement the needed algorithms or how to correctly use +cryptographic libraries. If knowledge of how to develop secure software is not +already established, the engineers have to acquire this knowledge. Management +has to provide funding to enable this type of security training (see also [9, sec. +$.l21). +If the training is not sufficient or the organization does not have adequatere- +sources for providing that kind of education, the organization has to acquire +external support. For instance, as the topic of PQC is currently under extensive +research, the organization may involve universities or institutes researching the +topic. +4.2 +Decide on time left +Based on Mosca's theorem [141, the senior management estimates the amount of +time left before CRQCs are available. This allows the development of a timeline +on which further decisions can be based. Of course, the decision has to be realistic +and based upon estimates by authorities like the BSI or the NIST. +4.3 +Security policy and goals +The PQC migration goals should be documented in the organization's goals and +need to be published inside the organization. Additionally, the security require- +ments for reaching the goals have to be documented in an organization-wide +security policy. Organizations applying an ISMS already have a security policy +that should be extended accordingly. +New software, either developed in-house or provided by third parties, can be +required to feature a certain level of cryptographic agility which can be used to +slater) implement PQC. +To set a migration timeline, the policy has to define the date on which CRQCs +are thought to be a real thing. As shown in Figure I, the security officer is in +charge of developing the security policy based on the management decisions. +The policy can feature concrete technical statements with which algorithms the +organization wants to protect which data. For instance, to exchange keys with a +party processing data requiring a medium level of security, KYBER-768 might +be chosen. In addition, the security policy has to document a list of excluded +algorithms if needed. This has to be coordinated with the communication part- +ners of the organization. Likewise, a list of preferred algorithms can be formed. +The decision is up to the organization because they know best which data has +which protection requirements. Alternatively, the security policy can reference a +technical document from the BSI or the NIST. +4.4 +Context of the organization +The context of the organization defines the scope in which cryptographic prim- +itives need to be upgraded. Also, the scope defines where the organization has + +PMMP +9 +to influence the upgrading of algorithmus. This Impacts the eomplexity of the +Migration and the mount of work needed. Below sections point out how the +context of the organization Can be defined and +hat it is Made of. In Figure 1, +this step is situated in the column of the executive officer Who is in Charge of +defining the context of the organization. +Stakeholders Stakeholders have an Interest in the organization"s success und +its conformity to laws, regulations, er contracts. Organizations have different +Stakeholders: Customers, employees, partners, suppliers, regulatory authorities, +a d themselves. As the success of an organization +an strongly depend on data +seeurity, the Stakeholders have an Interest in Information Security. +For instanz, an Interest in Information Security could be justified if the +organization handles Customer data er private Information of its partners. In +effet, the organization +an get obliged by its customers to migrate to PQC. +This Wright be the Case if a Service Level Agreement (SLA) exists between the +organization and its eustomers, defining the level of Security, availability, er +similar parameters defining the Service. +The Other Way +round: If the organization"s Service providers process data +that is relevant to the Stakeholders, it must be examined to what extent the +Service providers can be required by the organization to maintain the same level +of Security. Managing the migration to PQC in the organization would in this +Case also affect the suppliers of the organization, that Can get obliged to migrate +to PQC As weil. +If the organization is a regulated organization, like a Financial Institute, it +has to comply Voith certain l a s . Authorities Can require the organization to +fulfill legal er regulatory requirements as it is the esse for Financial Institutes. +Protection of Customer data Can be required by data proteetion l a s such as +the European General Data Protection Regulation (GDPR)3, which requires +organizations processing personal data to Secure data u sing State of the alt +techniques. Sorge day, State-of-the-art technology will include PQC and require +organizations to adapt their cryptography. +As the Stakeholders also have an economie Interest in the success of the +organization, they are required to agree upon the organization°s plans for migra- +tion [16, pp. l73-174]. This is because the migration requires financial resourees, +e.g, to provide upgraded hardware er to get external Support. +At last, Stakeholders can also be insurers Who Cover the organization against +cyber risks. If standardized PQC algorithmus are not used, the Insurance may be +invalidated. This Wright additionally be a driver of the migration. +Communication partners The Fisk quantum Computers pose apply especially +to public-key algorithmus like RSA. Public-key eryptography Systems are strongly +used for exchanging symmetric Keys er to develop a shared key. The algoríthms +are used to Secure data in transit. The data in transit is exchanged via insecure +3 See https://eur-lex.europa.eu/eli/reg/2016/679/oj +PMMP +9 +to influence the upgrading of algorithms. This impacts the complexity of the +migration and the amount of work needed. Below sections point out how the +context of the organization can be defined and what it is made of. In Figure 1, +this step is situated in the column of the executive officer who is in charge of +defining the context of the organization. +Stakeholders Stakeholders have an interest in the organization's success and +its conformity to laws, regulations, or contracts. Qrganizations have different +stakeholders: Customers, employees, partners, suppliers, regulatory authorities, +and themselves. As the success of an organization can strongly depend on data +security, the stakeholders have an interest in information security. +For instance, an interest in information security could be justified if the +organization handles customer data or private information of its partners. In +effect, the organization can get obliged by its customers to migrate to PQC. +This might be the case if a Service Level Agreement (SLA) exists between the +organization and its customers, defining the level of security, availability, or +similar parameters defining the service. +The other way around: If the organization's service providers process data +that is relevant to the stakeholders, it must be examined to what extent the +service providers can be required by the organization to maintain the same level +of security. Managing the migration to PQC in the organization would in this +case also affect the suppliers of the organization, that can get obliged to migrate +to PQC as well. +If the organization is a regulated organization, like a financial institute, it +has to comply with certain laws. Authorities can require the organization to +fulfill legal or regulatory requirements as it is the case for financial institutes. +Protection of customer data can be required by data protection laws such as +the European General Data Protection Regulation (GDpR)3, which requires +organizations processing personal data to secure data using state of the art +techniques. Some day, state-of-the-art technology will include PQC and require +organizations to adapt their cryptography. +As the stakeholders also have an economic interest in the success of the +organization, they are required to agree upon the organization's plans for migra- +tion [16, pp. 173-1741. This is because the migration requires financial resources, +e.g, to provide upgraded hardware or to get external support. +At last, stakeholders can also be insurers who cover the organization against +Cyber risks. If standardized PQC algorithms are not used, the insurance may be +invalidated. This might additionally be a driver of the migration. +Communication partners The risk quantum computers pose apply especially +to public-key algorithms like RSA. Public-key cryptography systems are strongly +used for exchanging symmetric keys or to develop a shared key. The algorithms +are used to secure data in transit. The data in transit is exchanged via insecure +3 See https://eur-lex.europa.eu/eli/reg/2016/679/oj + +10 +von Nethen et al. +Channels, like The Internet. Now, to Keep this data Secure, it is up to both side +to encrypt und decrypt the data sing the same algorithm. So, a successful +migration Can fly succeed if the Communication partners are taken long a d +also migrate; otherwise, there will be no Communication at all. +To prevent this fror happening, it is advised to not only compile an inventory +of cryptography as explained in Section 4.5, but also to compile an inventory +of Communication partners. The list of partners has to document +hat their +Communication endpoints Feature, e.g., which cryptographic primitive(s) they +are able to nun, in oder to exchange Information Voith the organization. +Additionally, the organization a d its communíeation partners have to agree +on an intersection of cryptographic primitives and parameters to ensure seeure +Communication after migration to PQC. Ideally, the algorithmus are supported +on the hardware the partners already are u sing +file fitting their needs. For +proper identification of the algorithmus, existing and standardized identifiers for +algorithmus have to be used. In the same manne, the partners should exclude +algorithmus they do not wart to use (because of trust issues and alike). +There are Communication partners Voith who +the organization Carnot agree +individually on algorithmus, e.g., Users of a Website. Depending on the need for +protection, these connections must be encrypted opportunistically. For this pur- +pose, the usual Users of the Website must be analyzed in oder to determine +which algorithmus Core i f o question. Then, they Can be provided Voith the best +possible Security. On a technieal layer, opportunistic Security is implemented +weithin the applications that are migrated. +With a so-called friends-and-family phase the migration of Selected appli- +cations Can be tested a d evaluated. There, only a Small number of Selected +Communication partners Upgrade their Systems to PQC. A successful prototyp- +ical migration +an her be used to advertise the use of PQC and get Other +Communication partners to migrate also. +Resources As noted in the previous section, the Management must provide the +resources for the migration, er., Investments in the education of the employees, +external Support, er Brewer hardware a d Software. Before money is spent, the +organization should assess +hat it Can Start without acquiring more resources. +That is, estimating which Computing resourees are used to which extent a d how +much room there is for algorithmus Voith higher resource demand. +Likewise, the resourees of the Software engineers need to be calculated. Are +they able to komplement new algorithmus er securely se Updates of cryptographic +libraries featuring new cryptographic primitives? In Figure l this relationship is +resembled by the dotted arrow connecting "Assess resources" in the rightmost +lane and the "Deine context of the organization" in the leftmost lane. +4.5 +Inventory of cryptography +To be üble to assess the Fisk a d the Impact of quantum Computing on the +organization, the organization needs to compile an inventory of eryptography +10 +von Nethen et al. +channels, like the internet. Now, to keep this data secure, it is up to both sides +to encrypt and decrypt the data using the same algorithm. So, a successful +migration can only succeed if the communication partners are taken along and +also migrate; otherwise, there will be no communication at all. +To prevent this from happening, it is advised to not only compile an inventory +of cryptography as explained in Section 4.5, but also to compile an inventory +of communication partners. The list of partners has to document what their +communication endpoints feature, e.g., which cryptographic primitive(s) they +are able to run, in order to exchange information with the organization. +Additionally, the organization and its communication partners have to agree +on an intersection of cryptographic primitives and parameters to ensure secure +communication after migration to PQC. Ideally, the algorithms are supported +on the hardware the partners already are using while fitting their needs. For +proper identification of the algorithms, existing and standardized identifiers for +algorithms have to be used. In the same manner, the partners should exclude +algorithms they do not want to use (because of trust issues and alike. +There are communication partners with whom the organization cannot agree +individually on algorithms, e.g., users of a website. Depending on the need for +protection, these connections must be encrypted opportunistically. For this pur- +pose, the usual users of the website must be analyzed in order to determine +which algorithms come into question. Then, they can be provided with the best +possible security. On a technical layer, opportunistic security is implemented +within the applications that are migrated. +With a so-called friends-and-family phase the migration of selected appli- +cations can be tested and evaluated. There, only a small number of selected +communication partners upgrade their systems to PQC. A successful prototyp- +ical migration can then be used to advertise the use of PQC and get other +communication partners to migrate also. +Resources As noted in the previous section, the management must provide the +resources for the migration, et., investments in the education of the employees, +external support, or newer hardware and software. Before money is spent, the +organization should assess what it can start without acquiring more resources. +That is, estimating which computing resources are used to which extent and how +much room there is for algorithms with higher resource demand. +Likewise, the resources of the software engineers need to be calculated. Are +they able to implement new algorithms or securely use updates of cryptographic +libraries featuring new cryptographic primitives? In Figure 1 this relationship is +resembled by the dotted arrow connecting "Assess resources" in the rightmost +lane and the "Define context of the organization" in the leftmost lane. +4.5 +Inventory of cryptography +To be able to assess the risk and the impact of quantum computing on the +organization, the organization needs to compile an inventory of cryptography + +PMMP +11 +(also called crypto-inventory). Risk management Can the decide which Systems +need to get migrated erst, based on their need for protection. White applying +this method, all relevant Systems must be discovered. This ensures the migration +process does not leave any Systems behind, undeteeted. The inventory provides +a list of applications +hat +se cryptography and also Shows which algorithmus +are used Voith which key lengths. This inventory serves as a starting point for +migration. In Figure 1, the creation of the crypto-inventory follows after defining +the context of the organization. This is the task of Security officers, s i r e they +has the resources to Interpret the technical details of the inventory and document +the Security level of the applications. +The inventory an be compiled u sing various methods. For instanz, if there +is few Systems, the inventory an be compiled by hand. Then, the administrators +of a System an be asked about the used cryptographic primitives. For bigger +environments, like a whole organization, Alnahawi et. al prognose the development +and usage of automated cryptography detection tools [1, p. 918]. These tools +could Scan Systems for, e.g., stored SSH Keys, libraries used in applications, +er trusted Foot certificates of a PKI. Applications that are developed in-house +by the organization and where it is possible to access the Source Code can be +scanned Voith text-based tools. For example, it Wright be sufficient to Scan the +Source Code for the terms encrypt er decrypt to und the used cryptographic +primitives. Furthermore, documents about formen Security audits and Interface +descriptions of The Systems Can help eompile a list of relevant Systems. +It Wright also be possible to Scan the +hole network traffic in an organization +to detect cryptography used in the network, er., TLS handshakes. This Works be- +cause TLS-protected applications (like HTTPS, LDAPS, FTPS, IMAPS, Open- +VPN, e t . ) +se the TLS handshake to negotiate the eipher Suite. Then, the +Chosen algorithmus a d key lengths could be recorded [15, pp. 9-l0]. Automated +tools are also recommended by NIST in [2, ll. 220-221]. However, there are still +no tools hat could be recommended at the moment. +PKIs need Special attention, as there are dependencies between Keys [8]. A +PKI can have many different entities and be rather complex. All entities in the +PKI have to migrate for the structure to function, as the certificates issued Wright +be installed on a large number of different devices. However, in a Way a PKI is +already same Kind of a crypto-inventory itself. +After the migration is done a d the applications are upgraded to PQC, the +cryptographic inventory is Updated. Then, the Fisk assessment can Start agar +Voith the most recht State of the inventory. +While eompiling the crypto-inventory, +it is important to doeument which +data is protected by the used algorithmus. To understand the application"s data +flow, reverse engineering of the System Wright be required. This helps assessing +the risks, as an be Seen in the next section. +4.6 +Assess risks +The developed process a r s to be Fisk-driven, which is why the Fisk Assessment +is on of The most important steps +her migrating efficiently und effectívely +PMMP +11 +also called crypto-inventoryl. Risk management can then decide which systems +need to get migrated first, based on their need for protection. While applying +this method, all relevant systems must be discovered. This ensures the migration +process does not leave any systems behind, undetected. The inventory provides +a list of applications that use cryptography and also shows which algorithms +are used with which key lengths. This inventory serves as a starting point for +migration. In Figure 1, the creation of the crypto-inventory follows after defining +the context of the organization. This is the task of security officers, since they +has the resources to interpret the technical details of the inventory and document +the security level of the applications. +The inventory can be compiled using various methods. For instance, if there +is few systems, the inventory can be compiled by hand. Then, the administrators +of a system can be asked about the used cryptographic primitives. For bigger +environments, like a whole organization, Alnahawi et. al propose the development +and usage of automated cryptography detection tools [1, p. 918]. These tools +could scan systems for, e.g., stored SSH keys, libraries used in applications, +or trusted root certificates of a PKI. Applications that are developed in-house +by the organization and where it is possible to access the source code can be +scanned with text-based tools. For example, it might be sufficient to scan the +source code for the terms encrypt or decrypt to find the used cryptographic +primitives. Furthermore, documents about former security audits and interface +descriptions of the systems can help compile a list of relevant systems. +It might also be possible to scan the whole network traffic in an organization +to detect cryptography used in the network, e.g., TLS handshakes. This works be- +cause TELS-protected applications (like HTTPS, LDAPS, FTPS, IMAPS, Open- +VPN, etc.) use the TLS handshake to negotiate the cipher suite. Then, the +chosen algorithms and key lengths could be recorded [15, pp. 9-I0]. Automated +tools are also recommended by NIST in [2, II. 220-221]. However, there are still +no tools that could be recommended at the moment. +PKIs need special attention, as there are dependencies between keys [81. A +PKI can have many different entities and be rather complex. All entities in the +PKI have to migrate for the structure to function, as the certificates issued might +be installed on a large number of different devices. However, in a way a PKI is +already some kind of a crypto-inventory itself. +After the migration is done and the applications are upgraded to PQC, the +cryptographic inventory is updated. Then, the risk assessment can start again +with the most recent state of the inventory. +While compiling the crypto-inventory, +it is important to document which +data is protected by the used algorithms. To understand the application's data +flow, reverse engineering of the system might be required. This helps assessing +the risks, as can be seen in the next section. +4.6 +Assess risks +The developed process aims to be risk-driven, which is why the risk assessment +is one of the most important steps when migrating efficiently and effectively + +12 +von Nethen et al. +to PQC. CRQCs Can pose a Fisk to the data Security of an organization. As +part of Fisk management, the Migration to PQC is considered a Fisk-minimizing +measure. This section explains the Interactions between Fisk management a d the +migration process and Shows how different risks should be handled. In Figure 1, +this step is situated in the lane of the Fisk ofñcer, which is responsible for the +Fisk assessment. The application developers ( a d maintainers) are responsible for +providing Information on the Systems that are subject to the Fisk assessment. +Therefore, there is an arrow connecting the toto rotes. +Supporting documents For each business process that is backed by different +applications the risks CRQCs pose need to be assessed. A business impaet anal- +ysis Can be helpful at this point. The Impact analysis Shows how the business is +affected in Case of interruptions to the Services provided by the IT Infrastructure +of the organization. Moreover, the Impact analysis helps to justify expenses for +the Migration. Additionally, existing documents on business continuation Can +help to assess the risks. With the additional help of a erypto-inventory, the ap- +plications at Fisk an be identified, s i r e the inventory Shows which applications +are sing algorithmus endangered by quantum Computers. Assessing the Fisk in- +volves evaluating how lang the data protected by the algorithmus used in the +application needs to stay seeure. If the time agreed upon by whieh CRQCs are +available is shorter han the time that is left to migrate,4 the Fisk is high and +the studied System has to be one of the first to get migrated. +Scope White applying the aforementioned techniques, that is, sing the resultat +of the business Impact analysis a d the inventory of cryptography, the Fisk as- +sessment has to over the +hole scope of the organization. This includes the +Stakeholders of the organization, which Wright be business partners, the organi- +zation itself, er authorities. +In the previous sections, it b e e r e Clear that organizations have different +Stakeholders. Not migrating to PQC Can have consequences that Stakeholders +Wright er Wright not wart to accept. For instanz, a Customer that fels their +data is no langer adequately protected could terminator the contract Voith the +organization er Ile a lawsuit. Authorities Wright impose severe penalties on the +organization. Furthermore, Communication partners that do not (wart to) mi- +grate to u sing PQC, Wright Fisk getting excluded fror the business. +Possible risks An attaeker Voith Access to a quantum Computer Wright deprive +the organization of its business. It is not possible to migrate all Systems at once. +For this reason, in same Cases, the risks must be accepted at least temporarily +before migrating. +4 Following The lines of +hat is knoten As Mosca"s theorem, the time that is left to +migrate to PQC is limited by the time needed to develop, prepare und deploy the +new scheue plus the time needed for secrets to refrain Secure. +12 +von Nethen et al. +to PQC. CRQCs can pose a risk to the data security of an organization. As +part of risk management, the migration to PQC is considered a risk-minimizing +measure. This section explains the interactions between risk management and the +migration process and shows how different risks should be handled. In Figure 1, +this step is situated in the lane of the risk officer, which is responsible for the +risk assessment. The application developers (and maintainers are responsible for +providing information on the systems that are subject to the risk assessment. +Therefore, there is an arrow connecting the two roles. +Supporting documents For each business process that is backed by different +applications the risks CRQCs pose need to be assessed. A business impact anal- +ysis can be helpful at this point. The impact analysis shows how the business is +affected in case of interruptions to the services provided by the IT infrastructure +of the organization. Moreover, the impact analysis helps to justify expenses for +the migration. Additionally, existing documents on business continuation can +help to assess the risks. With the additional help of a crypto-inventory, the ap- +plications at risk can be identified, since the inventory shows which applications +are using algorithms endangered by quantum computers. Assessing the risk in- +volves evaluating how long the data protected by the algorithms used in the +application needs to stay secure. If the time agreed upon by which CRQCs are +available is shorter than the time that is left to migrate,4 the risk is high and +the studied system has to be one of the first to get migrated. +Scope While applying the aforementioned techniques, that is, using the results +of the business impact analysis and the inventory of cryptography, the risk as- +sessment has to cover the whole scope of the organization. This includes the +stakeholders of the organization, which might he business partners, the organi- +zation itself, or authorities. +In the previous sections, it became clear that organizations have different +stakeholders. Not migrating to PQC can have consequences that stakeholders +might or might not want to accept. For instance, a customer that feels their +data is no longer adequately protected could terminate the contract with the +organization or file a lawsuit. Authorities might impose severe penalties on the +organization. Furthermore, communication partners that do not (want to) mi- +grate to using PQC, might risk getting excluded from the business. +Possible risks An attacker with access to a quantum computer might deprive +the organization of its business. It is not possible to migrate all systems at once. +For this reason, in some cases, the risks must be accepted at least temporarily +before migrating. +4 Following the lines of what is known as Mosca's theorem, the time that is left to +migrate to PQC is limited by the time needed to develop, prepare and deploy the +new scheme plus the time needed for secrets to remain secure. + +PMMP +13 +But, thinking of The performance requirements of the new algorithmus, there +is Brother Fisk: Sorge algorithmus providing a high level of Security Can Slow down +the Initial Connection to, e.g., a Website for up to many seconds. The organization +either has to accept the Fisk of losing eustomers beeause their Website appears to +be very Slow, er Upgrade the hardware at least on their Side of the Connection. +Also, it +an turn out that it is too hard to migrate an applieation so that a +replacement would be better. The post-quantum algorithmus may turn out to be +insecure during the migration. A required refurbishment er reverse-engineering +of an applieation Wright prove to be too eostly. +The organization°s Fisk management provides the deeision of whieh Systems to +migrate and when. Note that this deeision is eompletely Fisk-driven. For instanz, +Systems that operator in the inne network of the organization Wright not get +migrated erst as the Fisk is not high. Although they could be migrated Voith a +simple Upgrade in less than a few minutes, they handle unimportant data a d +are not reachable Over the Internet. +4.7 +Risk- and process-based grouping a d prioritization of +applications +Organizations use different applications to operator their business. The inter- +operability of the Systems must be ensured, s i r e any error may Interrupt the +organization7s processes. To prevent such issues, the applications and Systems +are grouped by the business process they are tied to. Then, the System Groups +are migrated on by one. In Figure l, the grouping of the applications is placed +in the lane of the Fisk officer. Preceding is the Fisk assessment of the Systems. +To recall the mitigation of the Y2K-Bug: It is imposant to remember that +the organization only has limited resources of Software engineers, which is why +Putnam and Schultz prognose to triage the Systems needing a migration [17, p. +96] [18, pp. 65-66]. First, the Systems that involve life a d death need to be +touched. Second, "if you are not in a life-and-death business" [17, p. 96], the +Systems Critical for running the organization°s business need to be worked on. +The Systems to be handled last are those +hose failure would be irritating, but +not costly. These Systems could be handled after l January 2000 (YZK). In oder +to adopt the process presented in [17] to PQC migration, the Systems need to +get triaged by their criticality regarding the data they process. +To conclude, the decision on which group is migrated erst is based on the +Fisk assessment done before. +4.8 +Testing a d monitoring +The Migration to PQC in an organization is controlled by the management. +It is a Fisk-driven process that is Integrated into the ISMS of an organization. +Software mígrations Can take quite a lang time. Therefore, the processes defined +in PMMP need to be monitored for effectiveness. Being a Cross layer aspect, the +testing a d monitoring step is not displayed In Figure l for better readability. It +PMMP +13 +But, thinking of the performance requirements of the new algorithms, there +is another risk: Some algorithms providing a high level of security can slow down +the initial connection to, e.g., a website for up to many seconds. The organization +either has to accept the risk of losing customers because their website appears to +be very slow, or upgrade the hardware at least on their side of the connection. +Also, it can turn out that it is too hard to migrate an application so that a +replacement would be better. The post-quantum algorithms may turn out to be +insecure during the migration. A required refurbishment or reverse-engineering +of an application might prove to be too costly. +The organization's risk management provides the decision of which systems to +migrate and when. Note that this decision is completely risk-driven. For instance, +systems that operate in the inner network of the organization might not get +migrated first as the risk is not high. Although they could be migrated with a +simple upgrade in less than a few minutes, they handle unimportant data and +are not reachable over the internet. +4.7 +Risk- and process-based grouping and prioritization of +applications +Organizations use different applications to operate their business. The inter- +operability of the systems must be ensured, since any error may interrupt the +organization's processes. To prevent such issues, the applications and systems +are grouped by the business process they are tied to. Then, the system groups +are migrated one by one. In Figure 1, the grouping of the applications is placed +in the lane of the risk officer. Preceding is the risk assessment of the systems. +To recall the mitigation of the YZK-Bug: It is important to remember that +the organization only has limited resources of software engineers, which is why +Putnam and Schultz propose to triage the systems needing a migration [17, p. +961 [18, pp. 65-661. First, the systems that involve life and death need to be +touched. Second, "if you are not in a life-and-death business" [17, p. 961, the +systems critical for running the organization's business need to be worked on. +The systems to be handled last are those whose failure would be irritating, but +not costly. These systems could be handled after 1 January 2000 (YQK). In order +to adopt the process presented in [17] to PQC migration, the systems need to +get triaged by their criticality regarding the data they process. +To conclude, the decision on which group is migrated first is based on the +risk assessment done before. +4.8 +Testing and monitoring +The migration to PQC in an organization is controlled by the management. +It is a risk-driven process that is integrated into the ISMS of an organization. +Software migrations can take quite a long time. Therefore, the processes defined +in PMMP need to be monitored for effectiveness. Being a cross layer aspect, the +testing and monitoring step is not displayed In Figure l for better readability. It + +14 +von Nethen et al. +is the Task of the Security officer to enforee the previously defined Security policy +(by ensuring the Migration processes do not top). +For instanz, regulated organizations like Financial Institutes that already +have an Internal control System in place ean use it to Monitor the effeetiveness +of the developed migration process. For example, a monitoring process Wright +Check if the migration processes defined an be applied correctly. The monitoring +has to detect a lack of resources required for migrating, e.g., lacking knowledge +in the Geld of PQC. In such a Case, the management has to improve resources +by organizing Workshops, establishing cooperation Voith universities, er releasing +more financial resources for the education of the developers. +To ensure that the methods a d techniques of PQC are applied correctly, +Security audits performed by external organizations +an help. Also, Internal se- +curity audits that the Security officer performs are helpful. +5 +Evaluation +In This Chapter, we verify that PMMP m e t s the defined requirements. Since +PMMP is based on the presence of a reestablished ISMS und its respective Change +processes the application of PMMP is possible wherever such a management +System is in place. Additionally, PMMP is compared to the migration approaches +presented in the literature. +5.1 Meeting the requirements +Migration Timeline PMMP Starts Voith educating Senior Management, to +ensure the topic is understood by the decision makers in the organization. Then, +the executive need to commit to a date by which CRQCs will be available a d +theater the currently used cryptography. Based on the Fisk assessment and Voith +the help of the inventory of cryptography, a migration timeline Can be defined. +To allow exeeutives to estimate the duration of the migration steps, PMMP +helps approximativ the mount of work needed. That is, the resources an orga- +nization has are estimated. Which application is getting migrated First is a risk- +based decision, not infiuencing the resources needed for the migration. U sing +techniques like reverse engineering to understand an application"s architecture, +the mount of work needed for the migration +an be assessed (for example by +measuring the complexity of the application +sing the number of lines of Code) . +Also, the migration process allows falling back to replacing the legacy applica- +tion Voith a new o n , if it is fester (er easier). To develop a migration timeline, +the process requires the User to Set the relevant dates stemming fror Mosca"s +theorem: The time when a CRQC will be available, the duration in which data +has to stay Secure, a d the duration of the migration to PQC. +Security PMMP Features the education of not +fly the decision makers but +also the education of, e.g., developers that komplement the new primitives. This +provides solid ground on which the algorithmus an be applied correetly. +14 +von Nethen et al. +is the task of the security officer to enforce the previously defined security policy +(by ensuring the migration processes do not stop). +For instance, regulated organizations like financial institutes that already +have an internal control system in place can use it to monitor the effectiveness +of the developed migration process. For example, a monitoring process might +check if the migration processes defined can be applied correctly. The monitoring +has to detect a lack of resources required for migrating, e.g., lacking knowledge +in the field of PQC. In such a case, the management has to improve resources +by organizing workshops, establishing cooperation with universities, or releasing +more financial resources for the education of the developers. +To ensure that the methods and techniques of PQC are applied correctly, +security audits performed by external organizations can help. Also, internal se- +curity audits that the security officer performs are helpful. +5 +Evaluation +In this chapter, we verify that PMMP meets the defined requirements. Since +PMMP is based on the presence of a reestablished ISMS and its respective change +processes the application of PMMP is possible wherever such a management +system is in place. Additionally, PMMP is compared to the migration approaches +presented in the literature. +5.1 Meeting the requirements +Migration Timeline PMMP starts with educating senior management, to +ensure the topic is understood by the decision makers in the organization. Then, +the executives need to commit to a date by which CRQCs will be available and +threaten the currently used cryptography. Based on the risk assessment and with +the help of the inventory of cryptography, a migration timeline can be defined. +To allow executives to estimate the duration of the migration steps, PMMP +helps approximate the amount of work needed. That is, the resources an orga- +nization has are estimated. Which application is getting migrated first is a risk- +based decision, not influencing the resources needed for the migration. Using +techniques like reverse engineering to understand an application's architecture, +the amount of work needed for the migration can be assessed (for example by +measuring the complexity of the application using the number of lines of code) . +Also, the migration process allows falling back to replacing the legacy applica- +tion with a new one, if it is faster (or easier. To develop a migration timeline, +the process requires the user to set the relevant dates stemming from Mosca's +theorem: The time when a CRQC will be available, the duration in which data +has to stay secure, and the duration of the migration to PQC. +Security PMMP features the education of not only the decision makers but +also the education of, e.g., developers that implement the new primitives. This +provides solid ground on which the algorithms can be applied correctly. + +PMMP +15 +Additionally, it is required that the process handles advancements in cryp- +tographic Analysis, which Gould weaken the Most promising PQC algorithmus. +PMMP solves this by integrating it into the Fisk Management of the organi- +zation. As s o n as one cryptographie primitive is weakened, the Situation is +evaluated agar, s i r e the Fisk management and the Fisk assessment are not a +one-shot process. Then, the migration process has to Start again Voith adapted +Security policies and algorithm selections. +Completeness PMMP makes se of the Features of an established ISMS, such +as System a d structure analysis. This is complemented by the creation of an +inventory of cryptography that serves, amor Other purposes, the compilation of +a list of migration-relevant Systems. Note that the formen a d the latte fackle +a Common task fror different angles, thereby complementing each Other. +Context awareness The migration process is required to be context-aware, +As +the Migration to PQC must be coordinated Voith business partners, to ensure +the partners Can communicate Voith each Other +file migrating. +PMMP solves this by involving business partners a d customers early in the +process, ensuring the relevant Communication partners have the same Vision. +Additionally, the process takes into account the needs of Stakeholders a d those +of regulatory authorities. +To ensure Communication partners of the organization are not exeluded +file +migrating, PMMP features a process to detect Communication partners, which +includes the primitives each partner Supports. +The selection of suitable algorithmus for different applieations is enabled by +complying Voith respective recommendations such as published by NIST er BSI. +Crypto-agility PMMP enables the establishment of erypto-agility by support- +ing the fulfillment of the requirements for practiced crypto-agility As defined in +the cryptographic agility maturity Model (CAMM) [10]. A More detailed pre- +sentation of the respective CAMM requirements ad how they are supported by +PMMP is g i e n in Appendix A. +Interoperability To prevent interrupting business processes, interoperabilíty +between the Systems is important. PMMP advises to se gateways between ex- +isting Systems. Then, Systems that are not yet upgraded to PQC Can connect +sing the gateway technology. +Also, Systems that are not able to be migrated, e.g., because the Software +vendors Carnot er do not wart to komplement PQC, er because the Source Code +of the Software Carnot be modified, can make use of gateways and ensure the +interoperabílíty of the Systems. +Additionally, the migration process is based on a process- and Fisk-based +groupíng of the applications a d Systems. Thereby, the interoperabílíty of the +Systems an be ensured. +PMMP +15 +Additionally, it is required that the process handles advancements in cryp- +tographic analysis, which could weaken the most promising PQC algorithms. +PMMP solves this by integrating it into the risk management of the organi- +zation. As soon as one cryptographic primitive is weakened, the situation is +evaluated again, since the risk management and the risk assessment are not a +one-shot process. Then, the migration process has to start again with adapted +security policies and algorithm selections. +Completeness PMMP makes use of the features of an established ISMS, such +as system and structure analysis. This is complemented by the creation of an +inventory of cryptography that serves, among other purposes, the compilation of +a list of migration-relevant systems. Note that the former and the latter tackle +a common task from different angles, thereby complementing each other. +Context awareness The migration process is required to be context-aware, as +the migration to PQC must be coordinated with business partners, to ensure +the partners can communicate with each other while migrating. +PMMP solves this by involving business partners and customers early in the +process, ensuring the relevant communication partners have the same vision. +Additionally, the process takes into account the needs of stakeholders and those +of regulatory authorities. +To ensure communication partners of the organization are not excluded while +migrating, PMMP features a process to detect communication partners, which +includes the primitives each partner supports. +The selection of suitable algorithms for different applications is enabled by +complying with respective recommendations such as published by NIST or BSI. +Crypto-agility PMMP enables the establishment of crypto-agility by support- +ing the fulfillment of the requirements for practiced crypto-agility as defined in +the cryptographic agility maturity model (CAMM) 1101. A more detailed pre- +sentation of the respective CAMM requirements ad how they are supported by +PMMP is given in Appendix A. +Interoperability To prevent interrupting business processes, interoperability +between the systems is important. PMMP advises to use gateways between ex- +isting systems. Then, systems that are not yet upgraded to PQC can connect +using the gateway technology. +Also, systems that are not able to be migrated, e.g., because the software +vendors cannot or do not want to implement PQC, or because the source code +of the software cannot be modified, can make use of gateways and ensure the +interoperability of the systems. +Additionally, the migration process is based on a process- and risk-based +grouping of the applications and systems. Thereby, the interoperability of the +systems can be ensured. + +16 +von Nethen et al. +Interim resultat Because the migration to PQC Can take a lang time, the +Migration process is required to deine a d deliver Interim resultat. Interim resultat +are delivered in various steps of the Migration. The erst step of the Migration is +to educate the executive management on the topic of quantum Computing and +its Fisk to classic cryptography. The knowledge gained in this step is an Interim +result. +Later in the migration, +her the migration strategy is formed, there exit dif- +ferent approaches for global migration. When the organization decides to takle +the migration +sing the incremental packet conversion approach, this method +delivers numerous Interim resultat. The application parts that are incrementally +put into produetion resemble Interim resultat. By sing gateway technologies it +can be ensured that the applications stay compatible Voith the rest of the orga- +nization. +Also, by rating awareness for the topic of PQC, a snowball effet an be +triggered. When Stakeholders of the organization are required to migrate to +PQC, this triggers stakeholder7s Stakeholders to migrate also. This is also an +intermediate result in the global migration to PQC. +Rea ct to advancements in cryptanalysis The Fisk Assessment of the organi- +zation is not a one-shot process. The Fisk is assessed regularly. Therefore, +her +advancements in cryptanalysis theater PQC er the knoten classic cryptographic +primitives, the migration process has to be applied agar. When advancements +are Made +file migrating, the Migration is adapted aceordingly. +5.2 +Comparison to existing migration approaches +One of the Most nature approaches existing is the on developed by Zhang +et. al, presented in [22]. What the approaches have in Common, is that the +Migration to PQC is triggered fror the top of the organization, initiated by the +decision makers that get edueated on the topic. The Second step is to eompile +an inventory of cryptography, but it is not mentioned how the inventory can +be compiled. Regarding the migration of on Single application (the IBM Db2 +database), the compilation of the inventory is not in their Focus. Moreover, the +toto approaches focus on the management of the migration. In [22], step Six +requires the User of the approach to execute the cybersecurity policy. The policy +Controls the selection of appropriate Solutions based on the requirements of the +organization and budgets. However, it refrains unclear how the requirements +can be determined. While the approaeh by Zhang et. al suggestiv Working on the +Systems that handle Critical data erst, it is not explained, what Critical data is. +Here, a Fisk-based process would be better suited, as proposed in this paper. +The migration approach presented in [12] Features a Fisk-based approach. +When the quantum Fisk for an organization is high, the approaeh suggestiv imple- +menting hybrid cryptography as sonn as possible. If the quantum Fisk is low, the +approach proposes to wart for an update for the application(s) in question. The +problem Voith this approach is that the Fisk assessment is done before eompiling +16 +von Nethen et al. +Interim results Because the migration to PQC can take a long time, the +migration process is required to define and deliver interim results. Interim results +are delivered in various steps of the migration. The first step of the migration is +to educate the executive management on the topic of quantum computing and +its risk to classic cryptography. The knowledge gained in this step is an interim +result. +Later in the migration, when the migration strategy is formed, there exist dif- +ferent approaches for global migration. When the organization decides to tackle +the migration using the incremental packet conversion approach, this method +delivers numerous interim results. The application parts that are incrementally +put into production resemble interim results. By using gateway technologies it +can be ensured that the applications stay compatible with the rest of the orga- +nization. +Also, by raising awareness for the topic of PQC, a snowball effect can be +triggered. When stakeholders of the organization are required to migrate to +PQC, this triggers stakeholder's stakeholders to migrate also. This is also an +intermediate result in the global migration to PQC. +React to advancements in cryptanalysis The risk assessment of the organi- +zation is not a one-shot process. The risk is assessed regularly. Therefore, when +advancements in cryptanalysis threaten PQC or the known classic cryptographic +primitives, the migration process has to be applied again. When advancements +are made while migrating, the migration is adapted accordingly. +5.2 +Comparison to existing migration approaches +One of the most mature approaches existing is the one developed by Zhang +et. al, presented in [22]. What the approaches have in common, is that the +migration to PQC is triggered from the top of the organization, initiated by the +decision makers that get educated on the topic. The second step is to compile +an inventory of cryptography, but it is not mentioned how the inventory can +be compiled. Regarding the migration of one single application (the IBM Db2 +databased, the compilation of the inventory is not in their focus. Moreover, the +two approaches focus on the management of the migration. In [221, step six +requires the user of the approach to execute the cybersecurity policy. The policy +controls the selection of appropriate solutions based on the requirements of the +organization and budgets. However, it remains unclear how the requirements +can be determined. While the approach by Zhang et. al suggests working on the +systems that handle critical data First, it is not explained, what critical data is. +Here, a risk-based process would be better suited, as proposed in this paper. +The migration approach presented in [12] features a risk-based approach. +When the quantum risk for an organization is high, the approach suggests imple- +menting hybrid cryptography as soon as possible. If the quantum risk is low, the +approach proposes to wait for an update for the application(s) in question. The +problem with this approach is that the risk assessment is done before compiling + +PMMP +17 +the crypto-inventory. It is unclear how The Fisk Can be assessed if it is unknown +which applications +se which type of cryptography, which key length und for +which data. PMMP puts Fisk management in Focus a d takes the advances in +quantum Computing as a Fisk that needs to be mitigated by the organization. +The basis for this process is the compilation of a crypto-inventory before as- +sessing the Fisk, s i r e the inventory is needed for the Fisk assessment. What the +approaches have in Common is that the approach presented in [12] also seeks to +get Support fror the decision makers in the organization. +One key differenz is that PMMP integrales into existing management pro- +cesses provided, e.g., by a nature Information Management Security System. +It uses the steering mechanismus in an organization for the migration to PQC +fror top to bottom. While applying the steps of PMMP Voith the testing and +monitoring in place, features of the Internal control System are used. +6 +Aehievements a d open issues +In the previous sections, PMMP a process for Managing the Migration to PQC +is presented. To develop a process hat is oriented to the needs of the industrie, +requirements were Set up. Then, the process was evaluated against these re- +quirements, to See how the migration process performs (in comparison to Other +approaches) ø +The successful evaluation of PMMP Shows that it can be used for migrating +fror classic cryptography to PQC. It is explained, which challenges have to be +solved a d how existing management methods +an be used in the context of +migrating to PQC. To achieve that, PMMP uses a Fisk-based method, regarding +the advances in quantum Computing as a Fisk that needs to be handled by Fisk +management. +Also, the process includes Interim States. When u sing the incremental packet +conversion approach, every application that is migrated to PQC and put into +production resembles an Interim State. Furthermore, the process puts a strong +emphasis on the context of the organization including its Stakeholders and com- +munication partners, Voith the latte berg one of the most important aspects of +a successful migration. This process Supports the cryptographie Understanding +of organizations: To be able to migrate to PQC, the organizations have to un- +derstand which applications are vulnerable a d what level of Security hey need +to provide for the data processed. +Although a migration process was developed, not every application Can be +migrated. There still exit applications that Carnot get migrated, e.g., due to +license reasons er because the Source ende is not available. This problem an be +solved sing a technique presented by Sneed et al. [9, sec. 4], i.e. sing gateways. +For instanz, a PQC-VPN can be used to ensure applications communicate +sing +PQC. +PMMP +17 +the crypto-inventory. It is unclear how the risk can be assessed if it is unknown +which applications use which type of cryptography, which key length and for +which data. PMMP puts risk management in focus and takes the advances in +quantum computing as a risk that needs to be mitigated by the organization. +The basis for this process is the compilation of a crypto-inventory before as- +sessing the risk, since the inventory is needed for the risk assessment. What the +approaches have in common is that the approach presented in [12] also seeks to +get support from the decision makers in the organization. +One key difference is that PMMP integrates into existing management pro- +cesses provided, e.g., by a mature Information Management Security System. +It uses the steering mechanisms in an organization for the migration to PQC +from top to bottom. While applying the steps of PMMP with the testing and +monitoring in place, features of the internal control system are used. +6 +Achievements and open issues +In the previous sections, PMMP a process for managing the migration to PQC +is presented. To develop a process that is oriented to the needs of the industry, +requirements were set up. Then, the process was evaluated against these re- +quirements, to see how the migration process performs (in comparison to other +approaches) • +The successful evaluation of PMMP shows that it can be used for migrating +from classic cryptography to PQC. It is explained, which challenges have to he +solved and how existing management methods can be used in the context of +migrating to PQC. To achieve that, PMMP uses a risk-based method, regarding +the advances in quantum computing as a risk that needs to be handled by risk +management. +Also, the process includes interim states. When using the incremental packet +conversion approach, every application that is migrated to PQC and put into +production resembles an interim state. Furthermore, the process puts a strong +emphasis on the context of the organization including its stakeholders and com- +munication partners, with the latter being one of the most important aspects of +a successful migration. This process supports the cryptographic understanding +of organizations: To he able to migrate to PQC, the organizations have to un- +derstand which applications are vulnerable and what level of security they need +to provide for the data processed. +Although a migration process was developed, not every application can be +migrated. There still exist applications that cannot get migrated, e.g., due to +license reasons or because the source code is not available. This problem can be +solved using a technique presented by Sneed et al. [9, sec. 4], i.e. using gateways. +For instance, a PQC-VPN can be used to ensure applications communicate using +PQC. + +18 +von Nethen et al. +6.1 +Open i +es +SS +Not all issues Gould be solved +file developing the Migration process. There +refrain a few open issues: +Earlier, it is mentioned +hat the organization has to assess the resources it +has to migrate to PQC. There e i s t resources that can easily be quantified, such +as the Financial backing er the amputation power of the Systems. But, also +developer resources need to be aecessed in oder to evaluate the need to provide +external Support (fror Other organizations er universities). If there is a metric, +that can be used to assess the eompetenee of Software engineers, it should be +used. +One of the most important tasks when migrating is to compile the inventory +of cryptography. When applying the developed migration process it may oeeur +that Systems are left out er overlooked a d do not get migrated to PQC. These +Systems pose a Security vulnerability. To the best of nur knowledge, to this +day, there is no conerete tool that can be recommended, as the researeh is still +ongoing. Nevertheless, due to the Fisk-based assessment of the Systems and the +cryptographic inventory, forgetting Systems is unlikely to happen Voith PMMP. +Further, if one System is left out in the migration, Other Systems may not be able +to connect to this System. Then, the overlooked System automatically receives +the attention. +In analogy to the development life Cycle of maturity models presented by +Becker et al. [3], the development of a migration process is an iterative process +that includes feedback fror outside experts a d practitioners. In the work at +hand we present nur erst proposal for PMMP to obtain external feedbaek. A +real-world migration has not yet been executed a d it refrains open how PMMP +will perform +her applied in practice. +7 +Conclusion +In This paper, the Migration to PQC was discussed fror different points of View. +A Management process for the migration was developed that is oriented to the +needs of the industrie. Throughout the paper, it was argued hat organizations +that wart to migrate to PQC have to Start as early as possible. The possible +threats fror advances in the Geld of quantum Computing Carnot be unseen. With +the developed migration process, a concept +hat helps fackle this Challenge was +formed. +An open question is how the open-souree Community will react to further +advances in quantum Computing. If IBM keeps its promises, powerful quantum +Computers will be cracking RSA sooner than later. Maybe a Worldwide Wave +of Innovation will bring PQC to many open-source products. Cryptography is +broadly used for securing traffic on the Internet, Voith HTTPS berg one major +use esse. As Google a d Cloudfiare showed, the devices of most Internet Users +are capable of running post-quantum Secure algorithmus. If Companies (like the +latte toto) use their large market power to initialize the migration to PQC, so +18 +von Nethen et al. +6.1 +Open i +u s +SS +Not all issues could be solved while developing the migration process. There +remain a few open issues: +Earlier, it is mentioned that the organization has to assess the resources it +has to migrate to PQC. There exist resources that can easily be quantified, such +as the financial backing or the computation power of the systems. But, also +developer resources need to be accessed in order to evaluate the need to provide +external support (from other organizations or universities). If there is a metric, +that can be used to assess the competence of software engineers, it should be +used. +One of the most important tasks when migrating is to compile the inventory +of cryptography. When applying the developed migration process it may occur +that systems are left out or overlooked and do not get migrated to PQC. These +systems pose a security vulnerability. To the best of our knowledge, to this +day, there is no concrete tool that can be recommended, as the research is still +ongoing. Nevertheless, due to the risk-based assessment of the systems and the +cryptographic inventory, forgetting systems is unlikely to happen with PMMP. +Further, if one system is left out in the migration, other systems may not be able +to connect to this system. Then, the overlooked system automatically receives +the attention. +In analogy to the development life cycle of maturity models presented by +Becker et al. [3], the development of a migration process is an iterative process +that includes feedback from outside experts and practitioners. In the work at +hand we present our first proposal for PMMP to obtain external feedback. A +real-world migration has not yet been executed and it remains open how PMMP +will perform when applied in practice. +7 +Conclusion +In this paper, the migration to PQC was discussed from different points of view. +A management process for the migration was developed that is oriented to the +needs of the industry. Throughout the paper, it was argued that organizations +that want to migrate to PQC have to start as early as possible. The possible +threats from advances in the field of quantum computing cannot be unseen. With +the developed migration process, a concept that helps tackle this challenge was +formed. +An open question is how the open-source community will react to further +advances in quantum computing. If IBM keeps its promises, powerful quantum +computers will be eraeking RSA sooner than later. Maybe a worldwide wave +of innovation will bring PQC to many open-source products. Cryptography is +broadly used for securing traffic on the internet, with HTTPS being one major +use ease. As Google and Cloudflare showed, the devices of most internet users +are capable of running post-quantum secure algorithms. If companies (like the +latter two use their large market power to initialize the migration to PQC, so + +PMMP +19 +that smaller organizations Wright feel the need to migrate, the global migration +Gould be sped up. Also, manufacturers of web browsers have strong power. For +example, the popular web browser Firefox displays a war fing ("connector not +seeure") when Users connect to an HTTP-only Site. The current Version of Firefox +also prevents connecting to a web Server sing the deprecated TLS 1.0 er 1.1 a d +displays a war fing. For an organization, e.g., a newspaper er an online Shop, this +Can result in a bad reputation, because the Users See an error when connecting +to the page. If one day quantum Computers are powerful enough, we may See +a similar war fing in web browsers we use then. Most promising Wright be an +update of widely used frameworks like OpenSSL to feature PQC. A new Version +of the framework that implements the new algorithmus Gould bring PQC to many +devices, as lang as they are powerful enough to nun the new primitives. +To get the migration on the Way, it is needed to repeat the very First step of +the migration process presented in this paper: Educate executives and decision +makers in PQC. Only if the possible dangers to the cryptography used today are +recognized, someone will Take the Money und Change Something. +Recalling the similarity of the Migration fror IPv4 to IPv6: there are net- +works that have already completely switched to sing the Brewer protocol, those +that se both variant, and those that still only se the oder variant. This devel- +opment will very likely also be Seen in the migration to PQC. In the beginning, +there will be only a few networks migrated to use post-quantum key exchanges +er encryption. In the meantime, there will be networks supporting both (er all +three) possible variants: classic cryptography, hybrid cryptography, a d PQC +only. +Also, NIST will most likely publish the Standards specifying PQC in the +next toto year (fror 2023). With these Standards, organizations can decide +more easily which algorithmus they need to komplement. Further work on the +topic Wright include u sing the developed migration process, improving it, a d +helping organizations migrate to PQC by increasing their cryptographic agility. +Meanwhile, we fall See, how the global migration to PQC will move Forward. +References +l. Alnahawi, N., Wiesmaier, A., Grasmeyer, T., Geißler, J., Zeier, A., Bauspieíš, +P., Heinemann, A.: On the State of post-quantum cryptography migration. 51. +Jahrestagung der Gesellschaft für Informatik, INFORMATIK +2021 - Computer +Science & Sustainability, Berlin, Germany, 27. September - l. Oktober, 2021 P- +314, 907-941 (2021). https://doi.org/10.18420/infolmatik2021-078, https : +//doi.org/10.18420/informatik2021-078 +2. Barker, W., Polk, W., Souppaya, M.: Getting Ready for Post-Quantum Cryp- +tography:: Explore Challenges Associated Voith Adoption +a d Use of Post- +Quantum Cryptographic +Algorithmus (May 2020). https://doi.org/10.6028/ +NIST.CSWP.05262020-draft, +https://nvlpubs.nist.gov/nistpubs/CSWP/NIST. +CSWP • 05262020-draf t .pdf +3. Becker, J., Knackstedt, R., Pöppelbulš, J.: Developing maturity models for +management. Bus. I n . Syst. Eng. +(3), 213-222 (06 2009) +1 +IT +PMMP +19 +that smaller organizations might feel the need to migrate, the global migration +could be sped up. Also, manufacturers of web browsers have strong power. For +example, the popular web browser Firefox displays a warning ("connection not +secure") when users connect to an HTTP-only site. The current version of Firefox +also prevents connecting to a web server using the deprecated TLS 1.0 or 1.1 and +displays a warning. For an organization, e.g., a newspaper or an online shop, this +can result in a bad reputation, because the users see an error when connecting +to the page. If one day quantum computers are powerful enough, we may see +a similar warning in web browsers we use then. Most promising might be an +update of widely used frameworks like OpenSSL to feature PQC. A new version +of the framework that implements the new algorithms could bring PQC to many +devices, as long as they are powerful enough to run the new primitives. +To get the migration on the way, it is needed to repeat the very first step of +the migration process presented in this paper: Educate executives and decision +makers in PQC. Only if the possible dangers to the cryptography used today are +recognized, someone will take the money and change something. +Recalling the similarity of the migration from IPv4 to IPv6: there are net- +works that have already completely switched to using the newer protocol, those +that use both variants, and those that still only use the older variant. This devel- +opment will very likely also be seen in the migration to PQC. In the beginning, +there will be only a few networks migrated to use post-quantum key exchanges +or encryption. In the meantime, there will be networks supporting both for all +three) possible variants: classic cryptography, hybrid cryptography, and PQC +only. +Also, NIST will most likely publish the standards specifying PQC in the +next two years (from 2023). With these standards, organizations can decide +more easily which algorithms they need to implement. Further work on the +topic might include using the developed migration process, improving it, and +helping organizations migrate to PQC by increasing their cryptographic agility. +Meanwhile, we shall see, how the global migration to PQC will move forward. +References +l. Alnahawi, N., Wiesmaier, A., Grasmeyer, T., Geiiéler, J., Zeier, A., Bauspielé, +P., Heinemann, A.: On the state of post-quantum cryptography migration. 51. +Jahrestagung der Gesellschaft fur Information, INFCRMATIK +2021 - Computer +Science & Sustainability, Berlin, Germany, 27. September - l. Oktober, 2021 P- +314, 907-941 (2021). https://doi.org/10.18420/infolmatik2021-078, https : +//doi.org/10.18420/informatik2021-078 +2. Barker, W., Polk, W., Souppaya, M.: Getting Ready for Post-Quantum Cryp- +tography:: Explore Challenges Associated with Adoption +and Use of Post- +Quantum Cryptographic +Algorithms +(May 2020). https://doi.org/10.6028/ +NIST.CSWP.05262020-draft, +https://nvlpubs.nist.gov/nistpubs/CSWP/NIST. +cswp . 05262020-draf t .pdf +3. Becker, J., Knackstedt, R., POppelbulé, J.: Developing maturity models for +management. Bus. Inf. Syst. Eng. +(3), 213-222 (06 2009) +1 +IT + +20 +von Nethen et al. +4. Bundesamt für Sicherheit +in der Informationstechnik +(BSI): Migration +zu +Post-Quanten-Kryptografie +- Handlungsempfehlungen +des +BSI +p. 9 (Aug +2020), +https://www.bsi.bund.de/SharedDocs/Downloads/DE/BSI/Krypto/ +Post-0uanten-Kryptografie.html +5. Bundesamt +für Sicherheit +in der Informationstechnik +(BSI): Kryptografie +quantensicher +gestalten. Tech. +Rep. +BSI-Bro21 Ol, Bundesamt +für Sicher- +heit +in der +Informationstechnik +(BSI), +Bonn +(Oct +2021), https://www . +bsi.bund.de/SharedDocs/Downloads/DE/BSI/Publikationen/Broschueren/ +Kryptografie-quantensicher-gestalten.pdf?__blob=publicationFile&v=4 +6. Computer Security Division, I.T.L.: Post-quantum cryptography | CSRC +y GSRC, +https://csrc.nist.gov/projects/post-quantum-cryptography +7. DIN Deutsches Institut +für Normung e. V.: German Version EN ISO IEC +2700I:2017. Beuth Verlag GmbH, Berlin (Jun 2017) +8. ETSI: Migration strategies a d recommendations to Quantum Safe scheues (Jul +2020),https://www.etsi.org/deliver/etsi_tr/103600_103699/103619/01.01. +01_60/tr_103619v010101p.pdf +9. Harry M. Sneed, Ellen Wolf, Heidi Heilmann: Softwaremigration +in der Praxis +ı Übertragung alter Softwaresysteme in eine moderne Umgebung. dpunkt.verlag, +Heidelberg, l edn. (2016) +10. Hoher, J., Heinemann, A., Wiesmaier, A.: Towards a maturity model for crypto- +agility assessment. In: l5th International Symposium on Foundations & Practice +of Security (FPS). Springer (2022) +11. ISARA Corporation: Managing Cryptographic a d Quantum Risk (Jul 2020) +12. Mashatan, A., Heintzman, D.: The Complex Path to Quantum Resistance: Is your +organization prepared? Queue 19(2), 65-92 (Apr 2021). https://doi.org/10 . +1145/3466132.3466779,https://dl.acm.org/doi/10.1145/3466132.3466779 +13. Michael Waidner, Ruben Niederhagen, Thorsten Grötker, Patrick Reinelt: Post- +Quantum +Crypto for dummes. +for dummes, WILEY-VCH Verlag GmbH +& Co. +KGaA, +Weinheim, +1 edn. +(2018), +https://www.utimaco.com/de/ +post-quantum-crypto-dummies +14. Mosca, +M.: +Cybersecurity +in +a +quantum +World: +will +we +be +ready? +Qåpr +2015 +https://csrc.nist.gov/csrc/media/events/ +workshop- on- cybersecurity- in- a-post - q_uantum-world/documents/ +presentations/session8-mosca-michele.pdf +15. Ott, D., Peikert, C., participants, o.w.: Identifying research challenge +in post quan- +tum cryptography Migration a d cryptographic agilíty (2019), http : I/arxiv . arg/ +abs/1909.07353 +16. Pandeya, G.R., Daim, T.U., Marotzke, A.: A Strategy Roadmap for Post-quantum +Cryptography. In: Daim, T.U. (ed.) Roadrnapping Future, pp. 171-207. Springer +International Publishing, Cham (2021), http://link.springer.com/10.1007/ +978-3-030-50502-8_4, Series Title: Applied Innovation and Technology Manage- +ment +17. Putnam, L., Myers, W.: Year 2000 work Comes down to the w i e 16 +96(1999).https://doi.org/10.1109/52.744575,http://ieeexplore.ieee.org/ +document/744575/ +18. Schultz, J.: Managing a y2k project-starting +now 15 +71 (1998). https:// +doi.org/10.1109/52.676742,http://ieeexplore.ieee.org/document/676742/ +19. Shor, PW.: Polynomial-Time +Algorithrns for Prime Factorization +and Discrete +Logarithmus on a Quantum Computer. SIAM Journal on Computing 26 +1509 «Der 1997).https://doi.org/10.1137/S0097539795293172,http://epubs. +siam.or8/doi/10.1137/S0097539795293172 +(3), 63- +(1), 90- +(5),l484- +20 +von Nethen et al. +4. Bundesamt fur Sicherheit +in der Informationstechnik +(BSI): Migration +zu +Post-Quanten-Kryptografie +- Handlungsempfehlungen +des +BSI +p. 9 (Aug +2020), +https://www.bsi.bund.de/SharedDocs/Downloads/DE/BSI/Krypto/ +Post-Quanten-Kryptografie.html +5. Bundesamt +fur Sicherheit +in der Informationstechnik +(BSI): Kryptografie +quantensicher +gestalten. Tech. +Rep. +BSI-Bro21 Ol, Bundesamt +fur Sicher- +heit +in der +Informationstechnik +(BSI), +Bonn +(Oct +2021), https://www . +bsi.bund.de/SharedDocs/Downloads/DE/BSI/Publikationen/Broschueren/ +Kryptografie-quantensicher-gestalten.pdf?__blob=publicationFile&v=4 +6. Computer Security Division, I.T.L.: Post-quantum cryptography | CSRC +y CSRC, +https://csrc.nist.gov/projects/post-quantum-cryptography +7. DIN Deutsches Institut +fur Nor rung e. V.: German version EN ISO IEC +27001:2017. Beuth Verlag GmbH, Berlin (Jun 2017) +8. ETSI: Migration strategies and recommendations to Qantum Safe schemes (Jul +2020),https://www.etsi.org/deliver/etsi_tr/103600_103699/103619/01.01. +01_60/tr_103619v010101p.pdf +9. Harry M. Sneed, Ellen Wolf, Heidi Heilmann: Softwaremigration +in der Praxis +• Ilbertragung alter Softwaresysteme in eine moder re Umgebung. dpunkt.verlag, +Heidelberg, l edit. (2016) +10. Hohm, J., Heinemann, A., Wiesmaier, A.: Towards a maturity model for crypto- +agility assessment. In: 15th International Symposium on Foundations & Practice +of Security (FPS). Springer (2022) +II. ISARA Corporation: Managing Cryptographic and Quantum Risk (Jul 2020) +12. Mashatan, A., Heintzman, D.: The Complex Path to Quantum Resistance: Is your +organization prepared? Queue 19(2), 65-92 (Apr 2021). https://doi.org/10 . +1145/3466132.3466779,https://d1.acm.org/doi/10.1145/3466132.3466779 +13. Michael Waidner, Ruben Niederhagen, Thorsten Grijtker, Patrick Reinelt: Post- +Quantum +Crypto for dummies. +for dummies, +WILEY-VCH +Verlag GmbH +& Co. +KCaA, +Weinheim, +I +edit. +(2018), +https://www.utimaco.com/de/ +post-quantum-crypto-dummies +14. Mosca, +M.: +Cybersecurity +in +a +quantum +world: +will +we +be +ready? +@Apr +2015 +https://csrc.nist.gov/csrc/media/events/ +workshop- on- cybersecurity- in- a-post - q_uantum-world/documents/ +presentations/session8-mosca-michele.pdf +15. Ctt, D., Peikert, C., participants, o.w.: Identifying research challenges in post quan- +tum cryptography migration and cryptographic agility (2019), http : I/arxiv . org/ +abs/1909.07353 +16. Pandeya, G.R., Daim, T.U., Marotzke, A.: A Strategy Roadmap for Post-quantum +Cryptography. In: Daim, T.U. (ed.) Roadmapping Future, pp. 171-207. Springer +International Publishing, Cham (2021), http://link.springer.com/10.1007/ +978-3-030-50502-8_4, series Title: Applied Innovation and Technology Manage- +ment +17. Putnam, L., Myers, W.: Year 2000 work comes down to the wire 16 +96(1999).https://doi.org/10.1109/52.744575,http://ieeexplore.ieee.org/ +document/744575/ +18. Schultz, J.: Managing a y2k project-starting +now 15 +71 (1998). https:// +doi.org/10.1109/52.676742,http://ieeexplore.ieee.org/document/676742/ +19. Shor, P.W.: Polynomial-Time +Algorithms for Prime Factorization +and Discrete +Logarithms on a Quantum Computer. SIAM Journal on Computing 26 +1509 UDct 1997).https://doi.org/10.1137/S0097539795293172,http://epubs. +siam.org/doi/10.1137/S0097539795293172 +(3), 63- +(1), 90- +(5),l484- + +PMMP +21 +20. TÜV Informationstechnik +GmbH: Whitepaper +Post-Quantum +Security (Nov +2020), +https://www.tuvit.de/en/innovations/post-quantum-cryptography/ +#c530188 +21. utimaco IS GmbH: Post-Quanten-Kryptografie: Sichere Verschlüsselung +für +das Quanten-Zeitalter (Mar 2018), https : I/www. infopoint-security.de/media/ +Utimaco_Whitepaper_0uantum-Computing_DE_vfinal.pdf +22. Zhang, L., Miranskyy, A., Rjaibi, W.: Quantum advantage a d the y2k bug: A +comparison. IEEE Software 38(2), 80-87 (2021). https://doi.org/10. 1109/MS . +2020.2985321, Conference Name: IEEE Software +23. Zhang, L., Miranskyy, A., Rjaibi, W., Stager, G., Gray, M., Peck, J.: Making Ex- +isting Software Quantum Safe: Lessons Learned. arXiv:2110.08661 [es] (Oct 2021), +http://arxiv.org/abs/2110.08661, archiv: 2110.08661 +A +CAl\/Il\/I requirements +PMMP m e t s the requirements of the cryptographic agility maturity model de- +veloped in [10] As show +in this section. White the CAMM requirements are +intended to measure (not establish) the crypto-agility of Systems, PMMP deliv- +ers processes +hose resultat fulfill the desired CAMM requirements up to CAMM +level 3 (practiced crypto-agility). We also take a look at CAMM level 4 require- +ments (sophisticated erypto-agility) und their relation to PMMP. +A.1 +CAl\/Il\/I level 1: "possible" +R1.0: System knowledge PMMP fulfills This requirement by involving stake- +holders of the organization a d compiling a list of Communication partners. By +that, the organization understands its context a d is able to evaluate the Impact +quantum Computers Wright have on it. PMMP Features a strong Focus on Fisk +management. +R1.1: Updateability PMMP fulfills This requirement by providing processes +Voith the Goal to Supply the needed resources for necessary Updates. For exam- +ple, the executive management is educated in cryptographic a d quantum Fisk. +Then, the management can provide monetary resourees to finanee the improving +process of crypto-agility. +To prevent restricting the functionality of Systems, their requirements need +to be Clear. Then, it an be measured if any functionality is lost. PMMP involves +reverse engineering processes where needed. +R1.2: Extensibility This requirement, in contrast to the requirement Update- +ability above, is not about assessing the ability to nun komplement PQC, but +actually about acquiring resources needed for the Upgrade. Resources have to +be approved to, er., buy new hardware, a d allow for external Support while +migrating cryptography. +PMMP +21 +20. TUV Informationsteohnik +GmbH: Whitepaper +Post-Quantum +Security (Nov +20201, +https://www.tuvit.de/en/innovations/post-quantum-cryptography/ +#0530188 +21. utimaco IS GmbH: Post-Quanten-Kryptografie: Sichere Verschliisselung +fur +das Quanten-Zeitalter (Mar 20181, https : I/www. infopoint-security.de/media/ +Utimaco_Whitepaper_0uantum-Computing_DE_vfinal.pdf +22. Zhang, L., Miranskyy, A., Rjaibi, W.: Quantum advantage and the y2k bug: A +comparison. IEEE Software 38(2), 80-87 (2021). https://doi.org/10. 1109/MS . +2020.2985321, conference Name: IEEE Software +23. Zhang, L., Miranskyy, A., Rjaibi, W., Stager, G., Gray, M., Peck, J.: Making Ex- +isting Software Quantum Safe: Lessons Learned. arXiv:2110.08661 [cs] (Oct 2021), +http://arxiv.org/abs/2110.08661, arXiv: 211008661 +A +CAl\/IM requirements +PMMP meets the requirements of the cryptographic agility maturity model de- +veloped in 1101 as shown in this section. While the CAMM requirements are +intended to measure (not establish) the crypto-agility of systems, PMMP deliv- +ers processes whose results fulfill the desired CAMM requirements up to CAMM +level 3 (practiced crypto-agility). We also take a look at CAMM level 4 require- +ments (sophisticated crypto-agility) and their relation to PMMP. +A.1 +CAl\/Il\/I level 1: "possible" +R1.0: System knowledge PMMP fulfills this requirement by involving stake- +holders of the organization and compiling a list of communication partners. By +that, the organization understands its context and is able to evaluate the impact +quantum computers might have on it. PMMP features a strong focus on risk +management. +R1.1: Updateability PMMP fulfills this requirement by providing processes +with the goal to supply the needed resources for necessary updates. For exam- +ple, the executive management is educated in cryptographic and quantum risk. +Then, the management can provide monetary resources to finance the improving +process of crypto-agility. +To prevent restricting the functionality of systems, their requirements need +to be clear. Then, it can be measured if any functionality is lost. PMMP involves +reverse engineering processes where needed. +R1.2: Extensibility This requirement, in contrast to the requirement Update- +ability above, is not about assessing the ability to run implement PQC, but +actually about acquiring resources needed for the upgrade. Resources have to +be approved to, e.g., buy new hardware, and allow for external support while +migrating cryptography. + +22 +von Nethen et al. +R1.3: Reversibility As the migration to PQC is managed per application, +Voith each application getting migrated in a dedicated project, this requirement +is met. Also, PMMP involves sing pilot Systems to ensure the Updates work as +expected. +R1.4: Cryptography inventory PMMP has defined processes for This re- +quirement. It deines how the inventory of cryptography +Can be compiled in +an organization. In conjunction Voith Fisk management, the level of Security the +cryptographic primitives provide a d which level of Security the data handled by +the applications where the primitives are used is understood. Plus, the process +compiles an inventory of Communication partners. +A.2 +CA1\/IM l +. +evel 2. +upr6p&r6dw +R2.0: Cryptographic modularity +tems of an organization are upgraded +PMMP ensures that applications a d sys- +in Groups based on their business process. +ID +R2.1: Algorithm +S +R2.2: Algorithm intersection +R2.3: Algo- +rithm exclusion PMMP has a strong focus on the context of the organization, +including analyzing its Stakeholders und Communication partners. In consulta- +tion Voith the Communication partners and especially their eapabilíties in the +changeover to PQC, PMMP enables a regulated changeover of the algorithmus. +The use of standardized algorithm identifiers is suggested for mentioned consul- +tation. In addition, the intersection and exclusion of algorithmus are determined +in the reconciliations Voith the Communication partners. Furthermore, Fisk-based +regulations can ensure that certain algorithmus are excluded, too. +R2.4: Opportunistic Security PMMP also deals Voith this issue fror The +background of the Communication partners. If no individual coordination Can +be made Voith a large number of partners, as is the Case, for example, Voith +the visitors to a globally accessible Website, the process relies on opportunistic +Security. For this purpose, the Users of the Systems are analyzed beforehand and +the best possible procedures are ensured. +A.3 +CAl\/Il\/I level 3: "practiced" +R3.0: Policies PMMP has a Great Focus on Security Management. For example, +the migration process initiales changes to an existing Security policy. The Security +policy is required to State +hat the organization warts in terms of migration to +PQC. The policy clarifies which requirements existing and Future applications +have to fulfill. Policies are developed in reconciliation Voith the Communication +partners, that Wright be required to also migrate. +22 +von Nethen et al. +R1.3: Reversibility As the migration to PQC is managed per application, +with each application getting migrated in a dedicated project, this requirement +is met. Also, PMMP involves using pilot systems to ensure the updates work as +expected. +R1.4: Cryptography inventory PMMP has defined processes for this re- +quirement. It defines how the inventory of cryptography can be compiled in +an organization. In conjunction with risk management, the level of security the +cryptographic primitives provide and which level of security the data handled by +the applications where the primitives are used is understood. Plus, the process +compiles an inventory of communication partners. +A.2 +CA1\/IM l +. +evel 2. +upr6p&r6dw +R2.0: Cryptographic modularity +rems of an organization are upgraded +PMMP ensures that applications and sys- +in groups based on their business process. +ID +R2.1: Algorithm +S +R2.2: Algorithm intersection +R2.3: Algo- +rithm exclusion PMMP has a strong focus on the context of the organization, +including analyzing its stakeholders and communication partners. In consulta- +tion with the communication partners and especially their capabilities in the +changeover to PQC, PMMP enables a regulated changeover of the algorithms. +The use of standardized algorithm identifiers is suggested for mentioned consul- +tation. In addition, the intersection and exclusion of algorithms are determined +in the reconciliations with the communication partners. Furthermore, risk-based +regulations can ensure that certain algorithms are excluded, too. +R2.4: Opportunistic security PMMP also deals with this issue from the +background of the communication partners. If no individual coordination can +be made with a large number of partners, as is the case, for example, with +the visitors to a globally accessible website, the process relies on opportunistic +security. For this purpose, the users of the systems are analyzed beforehand and +the best possible procedures are ensured. +A.3 +CAMM level 3: "practiced" +R.'8.0: Policies PMMP has a great focus on security management. For example, +the migration process initiates changes to an existing security policy. The security +policy is required to state what the organization wants in terms of migration to +PQC. The policy clarifies which requirements existing and future applications +have to fulfill. Policies are developed in reconciliation with the communication +partners, that might be required to also migrate. + +PMMP +23 +R3.1: Performance awareness PMMP takes into Account Security risks und +economic risks, such As losing customers that und themselves u sing a device +fly capable of performing a Slow post-quantum Secure TLS handshake. PMMP +deines the process of either accepting the risks er stocking up hardware, at least +on on Side of the Connection. +R3.2: Hardware modularity PMMP fulfills This application-specific require- +ment by including a refurbishment process in the migration process. For applica- +tions, for whieh the Fisk assessment revealed that they need to get migrated and +where the organization has the ability to influence the Software a d hardware, +making the needed +hanges to komplement PQC, the migration proeess ensures +that needed hanges are made +file migrating. +R3.3: Testing PMMP integrales into existing Management processes like Fisk +management. Because the Fisk assessment is not a one-shot process, it is regularly +checked whether the Security needs of the organization are fulfilled. +R3.4: Enforceability PMMP is drive by the leader of an organization, hat +is, the migration process is applied fror top to bottom. A Security policy put +in place by the organization°s executive management ensures that the needed +techniques, especially resources, are made available. Additionally, if resources +are lacking, e.g., educational resources, the migration process recommends co- +operating Voith universities and research institutions in the Geld. +R3.5: Security White PMMP is Applied, the Security of the organization is +assessed +sing external Security audits. To ensure the organization is not vulner- +able to attacks, Fisk Management assesses the seeurity of the organization. +R3.6: Backwards compatibility This requirement is fulfilled by PMMP +through its Focus on interoperabílity, which includes berg backward compat- +ible Voith oder Systems. Additionally, the process presents different management +strategies, e.g., the incremental packet conversion approach (fror [9]). With the +incremental packet conversion it is ensured, that the different parts of the ap- +plication migrated stay compatible Voith each Other. Moreover, the migration +process makes se of different transition mechanísms. +R3.7: Transition nıechanisnıs One technique that is used by PMMP is the +se of Gateways. This idem is taken fror the REMIP presented in [9]. Therefore, +PMMP Features processes to fulfill this requirement. +R3.8: Effectiveness PMMP makes +se of project Management techniques to +ensure upgrading cryptographíe primítíves is effective. Moreover, PMMP Can +PMMP +23 +R3.1: Performance awareness PMMP takes into account security risks and +economic risks, such as losing customers that find themselves using a device +only capable of performing a slow post-quantum secure TLS handshake. PMMP +defines the process of either accepting the risks or stocking up hardware, at least +on one side of the connection. +R3.2: Hardware modularity PMMP fulfills this application-specific require- +ment by including a refurbishment process in the migration process. For applica- +tions, for which the risk assessment revealed that they need to get migrated and +where the organization has the ability to influence the software and hardware, +making the needed changes to implement PQC, the migration process ensures +that needed changes are made while migrating. +R3.3: Testing PMMP integrates into existing management processes like risk +management. Because the risk assessment is not a one-shot process, it is regularly +checked whether the security needs of the organization are fulfilled. +R3.4: Enforceability PMMP is driven by the leaders of an organization, that +is, the migration process is applied from top to bottom. A security policy put +in place by the organization's executive management ensures that the needed +techniques, especially resources, are made available. Additionally, if resources +are lacking, e.g., educational resources, the migration process recommends co- +operating with universities and research institutions in the field. +R3.5: Security While PMMP is applied, the security of the organization is +assessed using external security audits. To ensure the organization is not vulner- +able to attacks, risk management assesses the security of the organization. +R3.6: Backwards compatibility This requirement is fulfilled by PMMP +through its focus on interoperability, which includes being backward compat- +ible with older systems. Additionally, the process presents different management +strategies, e.g., the incremental packet conversion approach (from 191l. With the +incremental packet conversion it is ensured, that the different parts of the ap- +plication migrated stay compatible with each other. Moreover, the migration +process makes use of different transition mechanisms. +R3.'7: Transition mechanisms One technique that is used by PMMP is the +use of gateways. This idea is taken from the REMIP presented in 191 Therefore, +PMMP features processes to fulfill this requirement. +R3.8: Effectiveness PMMP makes use of project management techniques to +ensure upgrading cryptographic primitives is effective. Moreover, PMMP can + +24 +von Nethen et al. +be monitored by the Internal control System. Theo, a lack of knowledge Can +be measured und reacted upon, for instanz. Additionally, PMMP features an +evaluation at the end of every application that is migrated. The Goal of this step +is to und ways to improve the Overall process, which ensures the effectiveness +of the Migration. Also, PMMP provides Solutions for Systems that Carnot get +upgraded in time ( a d have to be replaced Voith a Brewer application) . +A.4 +CAl\/Il\/I level 4: "sophisticated" +R4.0: Automation PMMP d e s not include processes for The Automation on +crypto-agility. Processes that could be included in the Migration are continuous +Integration a d continuous deployment pipelines that could be used to test the +applications. Then, +sing an automated process, the applications eould be auto- +matically checked whether they fulfill the requirements for crypto-agility. If not , +they could be automatically upgraded to PQC. +R4.1: Context Independence The techniques presented in PMMP are not +linked to a specific context. Rather, weithin the migration process, the eontext +of Migration is developed to match the relevant Scenario. Additionally, different +application architectures are considered in the process. Therefore, the migration +process Can be used in different contexts a d is context-independent. +R4.2: Scalability This requirement is not supported by PMMP. The migration +process has to be adapted to specific organizations a d Carnot be used for every +possible organization. +R4.3: Real-time PMMP includes processes that help Voith migrating in a g i e n +time. Moreover, there are techniques defined that strueture the Migration timely. +But, a Migration in real-time is not possible Voith PMMP. Techniques that could +allow a real-time migration are not considered. +R4.4: Interoperability Interoperability between various Systems is considered +in PMMP. For example, the se of gateways is considered. This allows applica- +tions that Carnot get upgraded to stay connected. Moreover, the process has a +strong focus on the context of the organization whieh ineludes its Stakeholders +(for example Users of a Website). +A.5 +Results +In the above evaluation against the requirements of CAMM, it is argued that +PMMP includes methods to Upgrade applications up to level 3 "practiced" of +CAMM. The requirements of level 4 cc sophistieated" are not all satisfied. +24 +von Nethen et al. +be monitored by the internal control system. Then, a lack of knowledge can +be measured and reacted upon, for instance. Additionally, PMMP features an +evaluation at the end of every application that is migrated. The goal of this step +is to find ways to improve the overall process, which ensures the effectiveness +of the migration. Also, PMMP provides solutions for systems that cannot get +upgraded in time (and have to be replaced with a newer application) . +A.4 +CAMM level 4: "sophisticated" +R4.0: Automation PMMP does not include processes for the automation of +crypto-agility. Processes that could be included in the migration are continuous +integration and continuous deployment pipelines that could be used to test the +applications. Then, using an automated process, the applications could be auto- +matically checked whether they fulfill the requirements for crypto-agility. If not , +they could be automatically upgraded to PQC. +R4.1: Context independence The techniques presented in PMMP are not +linked to a specific context. Rather, within the migration process, the context +of migration is developed to match the relevant scenario. Additionally, different +application architectures are considered in the process. Therefore, the migration +process can be used in different contexts and is context-independent. +R4.2: Scalability This requirement is not supported by PMMP. The migration +process has to be adapted to specific organizations and cannot be used for every +possible organization. +R4.3: Real-time PMMP includes processes that help with migrating in a given +time. Moreover, there are techniques defined that structure the migration timely. +But, a migration in real-time is not possible with PMMP. Techniques that could +allow a real-time migration are not considered. +R4.4: Interoperability Interoperability between various systems is considered +in PMMP. For example, the use of gateways is considered. This allows applica- +tions that cannot get upgraded to stay connected. Moreover, the process has a +strong focus on the context of the organization which includes its stakeholders +for example users of a website). +A.5 +Results +In the above evaluation against the requirements of CAMM, it is argued that +PMMP includes methods to upgrade applications up to level 3 "practiced" of +CAMM. The requirements of level 4 cc sophisticated" are not all satisfied. + diff --git a/MtE3T4oBgHgl3EQfYwqL/content/tmp_files/load_file.txt b/MtE3T4oBgHgl3EQfYwqL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..52d3df43333eb619e11bedaec38c035ec6cd05b6 --- /dev/null +++ b/MtE3T4oBgHgl3EQfYwqL/content/tmp_files/load_file.txt @@ -0,0 +1,1772 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf,len=1771 +page_content='Managing the Migration to Post-Quantum-Cryptography preprint - Nils von Nethen, Alex Wiesmaier, Oliver Weissmann, a d Nouri Alnahawi Darmstadt University of Applied Sciences, Germany Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=" Cryptographically relevant quantum Computers (CRQC) are presumably able to break today's prevalent classic cryptographic algo- rithms." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Protocols a d sehendes based on these algorithrns would become insecure if such CRQCs would become available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Although it is not ex- actly knoten, whether this will actually happen, organizations ( a d the IT Society) have to plan on migrating to quantum-resilient cryptographic measures, also knoten as Post-Quantum Cryptography (PQC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' However, migrating IT Systems a d applications in organizations to Support a d integrale new Software components is a difficult task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' There exists to the best of nur knowledge no generalized approach to manage such a complex migration for cryptography used in IT Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' We present a process for managing the migration fror classic cryptography to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Our Solution is based on best practices, challenges, a d problems derived fror established Software migration approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Compared to existing approaches, nur proposal provides a r e n s to help organizations migrate to PQC in a manageable manne a d maintain crypto-agility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Thus, nur process does not only serve as a framework for a one-time adaptation but also as a blueprint for organizing crypto-agile IT Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Keywords: Post-Quantum Cryptography (PQC) - PQC Migration Man- agement Process (PMMP) 1 Introduction Almosen all IT Systems und applications rely on cryptographic mechanismus to en- sure their Security against different types of attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Asymmetrie cryptographic scheues, such as RSA a d DH, have been always subject to threats fror ad- vances in cryptanalysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' These scheues are based on number theoretic hardness assumptions, which eould be broker, should one und efficient algorithmus for solv- ing t h e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Sufficiently large quantum Computers are assumed to pose such a Great threat, especially utilizing the light algorithmus [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Therefore, cryptographers This research work has been party funded by the German Federal Ministry of Edu- cation a d Research a d the Hessian State Ministry for Higher Education, Research a d the Arts weithin their joint Support of the National Research Center for Applied Cyber-Security ATHENE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Managing the Migration to Post-Quantum-Cryptography preprint - Nils von Nethen, Alex Wiesmaier, Oliver Weissmann, a d Nouri Alnahawi Darmstadt University of Applied Sciences, Germany Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=" Cryptographically relevant quantum computers (CRQC) are presumably able to break today's prevalent classic cryptographic algo- rithms." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Protocols and schemes based on these algorithms would become insecure if such CRQCs would become available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Although it is not ex- actly known, whether this will actually happen, organizations ( a d the IT society) have to plan on migrating to quantum-resilient cryptographic measures, also known as Post-Quantum Cryptography (PQC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' However, migrating IT systems a d applications in organizations to support a d integrate new software components is a difficult task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' There exists to the best of our knowledge no generalized approach to manage such a complex migration for cryptography used in IT systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' We present a process for managing the migration from classic cryptography to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Our solution is based on best practices, challenges, a d problems derived from established software migration approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Compared to existing approaches, our proposal provides a means to help organizations migrate to PQC in a manageable manner a d maintain crypto-agility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Thus, our process does not only serve as a framework for a one-time adaptation but also as a blueprint for organizing crypto-agile IT systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Keywords: Post-Quantum Cryptography (PQC) - PQC Migration Man- agement Process (PMMP) 1 Introduction Almost all IT systems a d applications rely on cryptographic mechanisms to en- sure their security against different types of attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Asymmetric cryptographic schemes, such as RSA a d DH, have been always subject to threats from ad- vances in cryptanalysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' These schemes are based on number theoretic hardness assumptions, which could be broken, should one find efficient algorithms for solv- ing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Sufficiently large quantum computers are assumed to pose such a great threat, especially utilizing the right algorithms 1191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Therefore, cryptographers This research work has been partly funded by the German Federal Ministry of Edu- cation a d Research a d the Hessian State Ministry for Higher Education, Research a d the Arts within their joint support of the National Research Center for Applied Cyber-Security ATHENE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 2 von Nethen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Worldwide have been, und are still, developing new eryptographic scheues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This Can be clearly Seen in the ongoing NIST PQC standardization process [6], the Goal of which is to establish additional Standard cryptographic scheues so that they can be Integrated into existing IT Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' However, adapting and migrat- ing large Software ínfrastructures to se PQC is an extremely difficult task hat is accompanied by several requirements a d challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In this paper, we present a process for managing the complex migration towards PQC in organizations a d IT Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Or methodologie observes this migration similar to any Other (emergency) Software migration process, such as in the Case of the famous Millennium Bug [22, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 80-85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Based on the related work presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 2, we erst identify the requirements and challenges for a suceessful migration, as weil as the needs and limitations of organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Building upon hat, a d upon existing migration approaches, we prognose a de- sign for nur migration management proeess (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 4), which also adopts same of the weil-studied and established concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Moreover, nur management process a r s at improving the cryptographic agility of an organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Therefore, we evaluate nur Solution based on the previously defined eriteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Thus, we ensure a process design that fits the needs of the industrie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In the following, nur process is abbrevíated Voith PMMP (PQC-Migration-Management-Process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 2 Related work We present related work grouped i f o (a) recommendations und overview papers and (b) existing management approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The formen Features ideas, challenges, a d governmental requirements for PQC migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The latte presents more er less concrete processes for conducting a migration towards PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='1 Recommendations and Overview NIST In an attempt to "Explore Challenges Associated Voith Adoption a d Use of Post-Quantum Cryptographic Algorithmus", the NIST Features the plan- ning of the Migration towards PQC in its White paper on "Getting Ready for Post-Quantum Cryptography" [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' However, this planning involves marly "ini- tial discovery steps for the development of migration roadmaps" [2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 6-7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In contrast to the context of this work, the NIST provide a broader View as they merton that the planning of the migration includes interacting Voith standards- developing organizations to reise awareness of necessary hanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To Start a migration Inside an organization they suggest prioritizing work by discovering Systems hat use public-key cryptography [2, lines 187-189] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Alnahawi et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' al The mount of research done on PQC is increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Alnahawi et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' al present a Survey that offers an overview on the work done so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' They de- scribe challenges as weil as already available Solutions for post-quantum-enabled protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' As a continuation of their printed publication, they nun a Website1 1 See https : I/pqc-cma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='io/cma/ 2 von Nethen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' worldwide have been, and are still, developing new cryptographic schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This can be clearly seen in the ongoing NIST PQC standardization process [61, the goal of which is to establish additional standard cryptographic schemes so that they can be integrated into existing IT systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' However, adapting and migrat- ing large software infrastructures to use PQC is an extremely difficult task that is accompanied by several requirements and challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In this paper, we present a process for managing the complex migration towards PQC in organizations and IT systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Or methodology observes this migration similar to any other (emergency) software migration process, such as in the case of the famous Millennium Bug [22, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 80-851.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Based on the related work presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 2, we first identify the requirements and challenges for a successful migration, as well as the needs and limitations of organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Building upon that, and upon existing migration approaches, we propose a de- sign for our migration management process (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' ill, which also adopts some of the well-studied and established concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Moreover, our management process aims at improving the cryptographic agility of an organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Therefore, we evaluate our solution based on the previously defined criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Thus, we ensure a process design that fits the needs of the industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In the following, our process is abbreviated with PMMP (PQC-Migration-Management-Process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 2 Related work We present related work grouped into (al recommendations and overview papers and (be existing management approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The former features ideas, challenges, and governmental requirements for PQC migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The latter presents more or less concrete processes for conducting a migration towards PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='1 Recommendations and Overview NIST In an attempt to "Explore Challenges Associated with Adoption and Use of Post-Quantum Cryptographic Algorithms", the NIST features the plan- ning of the migration towards PQC in its white paper on "Getting Ready for Post-Quantum Cryptography" 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' However, this planning involves mainly "ini- tial discovery steps for the development of migration roadmaps" [2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 6-7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In contrast to the context of this work, the NIST provide a broader view as they mention that the planning of the migration includes interacting with standards- developing organizations to raise awareness of necessary changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To start a migration inside an organization they suggest prioritizing work by discovering systems that use public-key cryptography [2, lines 187-1891 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Alnahawi et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' al The amount of research done on PQC is increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Alnahawi et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' al present a survey that offers an overview on the work done so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' They de- scribe challenges as well as already available solutions for post-quantum-enabled protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' As a continuation of their printed publication, they run a website1 1 See https : I/pqc-cma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='io/cma/ PMMP 3 hat organizes the State of the Art and is intended to possibly refrain up-to-date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The Website and íts reference were extensívely used file Writing the paper at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' BSI The German Federal Office for Information Security (BSI) provides recom- mendations for migrating so PQC in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The Institute suggestiv enabling crypto- agility to react to changing Security levels of the used cryptography, but it is not explained how to achieve the needed levels of crypto-agility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' However, they emphasize their position by reeommending the use of hybrid Solutions [4] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' ISARA In [11], the migration to PQC is compared to the migration needed for the Y2k-bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The migration is drive by Fisk management, thus involving the leaders of organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' They explicitly State that advanees in erypto-analysis theater classic cryptography [11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 7-l5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The erst step hey prognose is to increase the eryptographic agility by increasing cryptographic visibility [11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, they recommend including cryptographic and quantum crypto- graphic risks in the cyber Fisk strategy of an organization [11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Their approach distinguishes between the quantum Fisk [11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' l6-23] and the eryptographie Fisk [11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 7-15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To manage the quantum Fisk, the organiza- tion needs to become crypto-agile [11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 23], which is quite similar to managing the eryptographic Fisk as a hole, but rather focusing on the speeific needs of PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='2 Existing management approaches Zhang et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' al In [23], the lessons learned fror migrating an IBM Db2 database to sing PQC are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The approach, explained in detail in [22, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 85-86], is inspired fror the migrations mitigating the Y2K-bug [22, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Zhang et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' al State that organizations Can Start planning now a d make Sure hey are prepared, by investing in cryptographic agility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' They prognose to se Software design patterns such as the Factory pattern to e fable replacing crypto- graphic primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Considering the problem of coordínating Voith business part- ners, the authors refer to the IETF, which is already Working on new Internet Standards featuring PQC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' They State that the task of migrating to PQC is a Community task a d that Software practítioners all Over the World have to handle, espeeially by upgrading the Software they maintain to be crypto-agile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' One of the hardest challenge they eneountered while migrating the IBM Db2 database was the lack of quality of documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Most documents were out of date so they offen needed to reverse-engineer the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, they offen encountered hard-coded key lengths that needed to be Updated [23, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In the end, the migration was sueeessful Voith even slightly better response times [23, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 2 mea https://trac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='ietf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='org/trac/sec/wiki/PQCAgility PMMP 3 that organizes the state of the art and is intended to possibly remain up-to-date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The website and its references were extensively used while writing the paper at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' BSI The German Federal Office for Information Security (BSI) provides recom- mendations for migrating to PQC in 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The institute suggests enabling crypto- agility to react to changing security levels of the used cryptography, but it is not explained how to achieve the needed levels of crypto-agility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' However, they emphasize their position by recommending the use of hybrid solutions 141 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' ISARA In [11], the migration to PQC is compared to the migration needed for the Y2k-bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The migration is driven by risk management, thus involving the leaders of organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' They explicitly state that advances in crypto-analysis threaten classic cryptography 111, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 7-l5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The first step they propose is to increase the cryptographic agility by increasing cryptographic visibility 111, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, they recommend including cryptographic and quantum crypto- graphic risks in the Cyber risk strategy of an organization 111, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Their approach distinguishes between the quantum risk [11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 16-23] and the cryptographic risk 111, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 7-15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To manage the quantum risk, the organiza- tion needs to become crypto-agile [11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 23], which is quite similar to managing the cryptographic risk as a whole, but rather focusing on the specific needs of PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='2 Existing management approaches Zhang et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' al In [23], the lessons learned from migrating an IBM Db2 database to using PQC are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The approach, explained in detail in [22, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 85-861, is inspired from the migrations mitigating the YAK-bug [22, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 841.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Zhang et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' al state that organizations can start planning now and make sure they are prepared, by investing in cryptographic agility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' They propose to use software design patterns such as the factory pattern to enable replacing crypto- graphic primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Considering the problem of coordinating with business part- ners, the authors refer to the IETF, which is already working on new internet standards featuring poo?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' They state that the task of migrating to PQC is a community task and that software practitioners all over the world have to handle, especially by upgrading the software they maintain to be crypto-agile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' One of the hardest challenges they encountered while migrating the IBM I)b2 database was the lack of quality of documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Most documents were out of date so they often needed to reverse-engineer the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, they often encountered hard-coded key lengths that needed to be updated [23, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In the end, the migration was successful with even slightly better response times [23, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' I2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 2 mea https://trac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='ietf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='org/trac/sec/wiki/PQCAgility 4 von Nethen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' la/Iashatan a d Heinzmann In [12], three different pathos are presented for migrating to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The authors advise to establish a governance model und body to Follow their recommendations for migrating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Following that, they ad- vise to assess the risks quantum Computers Wright pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This is done by exam- ining the current cryptographic footprint of the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Afterwards, the quantum-resistant alternatives should be Selected and implemented according to the Chosen path [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' All pathos lead to remediation projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The authors describe this task as implementing the newly published eryptography Standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Organizations that have Chosen to follow path A will the observe their position in the Geld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Orga- nizations following path B will the execute their roadmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' And organizations following path C will the "simply need to make (relatively minor) adjustments to hat they already have in place" [12, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, the authors suggest the organizations will also need to have a deprecation path for the pre-quantum cryptographic implementations in plaee [12, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 3 Requirements In This section, the requirements of the developed Migration Management ap- proach to PQC are described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Defining these requirements is mandatory for the evaluation afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Problems defined in existing White papers er migration drafts, as weil as lessons learned fror the migrations presented above [12,23] are used as a basis for the requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Migration tinıeline As stated in [20, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 20], the Integration of PQC Can a d should Start today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In [11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 23], it is Generally recommended to Start early, s i r e cryptographic upgrades are challenging and time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The Utimaco GmbH similarly suggestiv putting the topic on the agenda of the organization, in oder to prevent harvest-then-decrypt attaeks [21, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Furthermore, in [22, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 16], the authors advise having a Clear roadmap a d a timeline for the migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Consequently, the PMMP needs to help deine a timeline for the migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' It has to be possible for executives to estimate the duration of each step a d the migration as a hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Therefore, each phase of the migration has to have metrics by which the duration an be measured, estimated, a d steered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Security The Migration process needs to ensure hat a System uses post- quantum Secure algorithmus afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This m a n s that the algorithmus imple- mented need to mitigate the Fisk of an attacke utilizingg a CRQC to decrypt the organization°s Communication Voith respect to the lifetime of the secret data, er faking authentication data er digital signatures .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The migration process itself must not allow for new vulnerabilities to open up file it is berg executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For instanz, disabling cryptography modules completely, because hey Carnot get migrated, must not be an Option a d must be prevented by the proeess Voith appropriate countermeasures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Applications 4 von Nethen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Mashatan and I-Ieinzmann In [121, three different paths are presented for migrating to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The authors advise to establish a governance model and body to follow their recommendations for migrating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Following that, they ad- vise to assess the risks quantum computers might pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This is done by exam- ining the current cryptographic footprint of the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Afterwards, the quantum-resistant alternatives should be selected and implemented according to the chosen path [121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' All paths lead to remediation projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The authors describe this task as implementing the newly published cryptography standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Organizations that have chosen to follow path A will then observe their position in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Orga- nizations following path B will then execute their roadmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' And organizations following path C will then "simply need to make (relatively minors adjustments to what they already have in place" [12, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, the authors suggest the organizations will also need to have a deprecation path for the pre-quantum cryptographic implementations in place [12, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 3 Requirements In this section, the requirements of the developed migration management ap- proach to PQC are described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' DeNning these requirements is mandatory for the evaluation afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Problems defined in existing white papers or migration drafts, as well as lessons learned from the migrations presented above 112,231 are used as a basis for the requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Migration timeline As stated in [20, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 20], the integration of PQC can and should start today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In 111, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 23], it is generally recommended to start early, since cryptographic upgrades are challenging and time-eonsuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The Utimaco GmbH similarly suggests putting the topic on the agenda of the organization, in order to prevent harvest-then-decrypt attacks [21, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' ill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Furthermore, in 122, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 16], the authors advise having a clear roadmap and a timeline for the migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Consequently, the PMMP needs to help define a timeline for the migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' It has to be possible for executives to estimate the duration of each step and the migration as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Therefore, each phase of the migration has to have metrics by which the duration can be measured, estimated, and steered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Security The migration process needs to ensure that a system uses post- quantum secure algorithms afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=" This means that the algorithms imple- mented need to mitigate the risk of an attacker utilizingg a CRQC to decrypt the organization's communication with respect to the lifetime of the secret data, or faking authentication data or digital signatures ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The migration process itself must not allow for new vulnerabilities to open up while it is being executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For instance, disabling cryptography modules completely, because they cannot get migrated, must not be an option and must be prevented by the process with appropriate countermeasures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Applications PMMP 5 offering both, classic cryptography und PQC must not allow downgrade attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' If the Communication Voith a partner is changed to sing PQC, ít must not revert to legacy algorithmus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Exceptions will be needed, but have to be reviewed a d accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Moreover, the newly implemented post-quantum Secure algorithmus have to be used correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For example, the process Wright provide processes for educating participants in the migration (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', programmes, administrators, er project leaders) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Furthermore, the process needs to handle advancements in cryptographic analysis, weakening formerly promising PQC algorithmus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In [20, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 21], it is rec- ommended to se hybrid methods that se pre- and post-quantum cryptography simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Hybrid methods are also used in the process presented in [11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 23], in Order to minimize the Fisk of harvest-and-decrypt attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' They prognose that the process must favor hybrid methods for higher-risk applications a d pro- mote cryptographie agility in the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In [13, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 41], it is suggested to "reise hybrids", by which they mea implementing hybrid algorithmus, supporting post-quantum and classic primitives at the same time, thus taking the best of both worlds and enabling interoperability between Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Therefore, it an be assumed that hybrid Solutions will probably become the de facto Standard Way of integrating PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Completeness The Migration process has to ensure that all relevant Systems that need a migration to PQC actually get migrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The migration process has to recommend a mechanisch by which a comprehensive list of Systems needing a migration an be compiled [22, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Context awareness In [11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 24], it is recommended to ask vendors of cryp- tographie products about their "quantum-readiness".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For the migration process, this m a n s that on needs mechanismus that assess the cryptographic agility of vendors if third-party Systems are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Most importantly, the migration to PQC Can only sueceed if Communication partners migrate as weil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The migration pro- cess is required to ensure hat a context-aware Migration strategy is developed a d applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, in [23, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 14], the authors report a steep learning curve for devel- opers wanting to integrale PQC, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' the need to acquire substantiell knowledge on the topic before starting the actual work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To conquer this Challenge, it is re- quired to ensure that Software erıgineers involved in the migration are educated weil enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In [16] it is pointed out, that employees have to be educated in the new technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The migration process has to elaborate on how to educate workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For instanz, it is suggested to collaborate Voith universities, as these provide a large pool of talents [16, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 199].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This eould be used for the edueation of the employees a d for helping Voith the migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The different post-quantum Secure algorithmus are suitable for different needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Approaching a new er in cryptography, there will no langer be on universal cryptography that Can be used for all se Cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' As required in [20, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 22], the migration has to be tailored to the needed Security level of the protected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' PMMP 5 offering both, classic cryptography and PQC must not allow downgrade attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' If the communication with a partner is changed to using PQC, it must not revert to legacy algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Exceptions will be needed, but have to he reviewed and accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Moreover, the newly implemented post-quantum secure algorithms have to be used correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For example, the process might provide processes for educating participants in the migration (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', programmers, administrators, or project leaders) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Furthermore, the process needs to handle advancements in cryptographic analysis, weakening formerly promising PQC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In [20, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 211, it is rec- ommended to use hybrid methods that use pre- and post-quantum cryptography simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Hybrid methods are also used in the process presented in [11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 23], in order to minimize the risk of harvest-and-decrypt attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' They propose that the process must favor hybrid methods for higher-risk applications and pro- mote cryptographic agility in the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In [13, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 411, it is suggested to "raise hybrids", by which they mean implementing hybrid algorithms, supporting post-quantum and classic primitives at the same time, thus taking the best of both worlds and enabling interoperability between systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Therefore, it can be assumed that hybrid solutions will probably become the de facto standard way of integrating PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Completeness The migration process has to ensure that all relevant systems that need a migration to PQC actually get migrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The migration process has to recommend a mechanism by which a comprehensive list of systems needing a migration can be compiled 122, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Context awareness In [11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 241, it is recommended to ask vendors of cryp- tographic products about their "quantum-readiness".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For the migration process, this means that one needs mechanisms that assess the cryptographic agility of vendors if third-party systems are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Most importantly, the migration to PQC can only succeed if communication partners migrate as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The migration pro- cess is required to ensure that a context-aware migration strategy is developed and applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, in [23, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 141, the authors report a steep learning curve for devel- opers wanting to integrate PQC, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' the need to acquire substantial knowledge on the topic before starting the actual work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To conquer this challenge, it is re- quired to ensure that software engineers involved in the migration are educated well enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In 1161 it is pointed out, that employees have to be educated in the new technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The migration process has to elaborate on how to educate workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For instance, it is suggested to collaborate with universities, as these provide a large pool of talents [16, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This could be used for the education of the employees and for helping with the migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The different post-quantum secure algorithms are suitable for different needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Approaching a new era in cryptography, there will no longer be one universal cryptography that can be used for all use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' As required in [20, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 22], the migration has to be tailored to the needed security level of the protected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 6 von Nethen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The decision on which algorithm is used has to be balaneed between Speed, s i e , und Security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Consequently, the Migration process has to Guide the selection of suitable algorithmus vor different applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Crypto-agility Technical debüts, such as hard-coded key lengths, add further levels of complexity [22, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The BSI recommends implementing crypto- graphic agility file new applications are developed er existing ortes are up- graded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This enables the organization to update the cryptographic primitives More easily [5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Other organizations, such as ISARA [11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 24] er uti- maco [21, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 5-6] also recommend this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Thus, the Migration process must present mechanismus that an help Voith ensuring a sufficient level of cryptographie agility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Furthermore, the BSI recommends hybrid Solutions that combine classic cryptography Voith post-quantum Solutions [5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For high Security areas, the BSI even requires hybrid Solutions [4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 7] Interoperability a d availability White migrating an application, connected applications need to stay able to communicate Voith the migrated application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Interoperabílity needs to be ensured for the organization as a hole and must not Interrupt the Operation of the organization°s business processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' As an example, in [11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 9], it is required to komplement "Forward backward compatibility" such that Systems an still operator during migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Interim resultat Migrating a hole organization fror sing classic cryptogra- phy to PQC is a process that Can take quite a file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The process must be able to deliver Interim resultat that Can be used before the entre organization is mi- grated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' An example of an Interim result is a Single application that is migrated to PQC (but is still able to communicate Voith connected applications that have not yet been migrated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Rea ct to advancements in cryptanalysis If there are advancements in cryptanalysis, The process has so provide Solutions that Can be Applied her the implemented algorithmus turn out to be insecure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This is very related to the before-mentioned crypto-agílíty requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 4 Defining the la/Ianagement Process In This section, we deine nur management process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In the context of this paper migration r e n s replacing eryptographic primitives Voith those that Can protect data fror an attacker Voith access to a quantum Computer powerful enough to break classic cryptography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To be able to aequire the resources, approval by the decision makers of the organization is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The migration to PQC is a complex task that requires substantiell resources, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' more powerful hardware, time, er external Support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In 6 von Nethen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The decision on which algorithm is used has to he balanced between speed, size, and security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Consequently, the migration process has to guide the selection of suitable algorithms for different applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Crypto-agility Technical debts, such as hard-coded key lengths, add further levels of complexity 122, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The BSI recommends implementing crypto- graphic agility while new applications are developed or existing ones are up- graded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This enables the organization to update the cryptographic primitives more easily 15, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Other organizations, such as ISARA 111, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 241 or uti- maco 121, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 5-61 also recommend this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Thus, the migration process must present mechanisms that can help with ensuring a suftieient level of cryptographic agility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Furthermore, the BSI recommends hybrid solutions that combine classic cryptography with post-quantum solutions 15, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For high security areas, the BSI even requires hybrid solutions 14, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 71 Interoperability and availability While migrating an application, connected applications need to stay able to communicate with the migrated application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=" Interoperability needs to be ensured for the organization as a whole and must not interrupt the operation of the organization's business processes." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' As an example, in [11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 91, it is required to implement "forward backward compatibility" such that systems can still operate during migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Interim results Migrating a whole organization from using classic cryptogra- phy to PQC is a process that can take quite a while.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The process must be able to deliver interim results that can be used before the entire organization is mi- grated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' An example of an interim result is a single application that is migrated to PQC (but is still able to communicate with connected applications that have not yet been migrated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' React to advancements in cryptanalysis If there are advancements in cryptanalysis, the process has to provide solutions that can be applied when the implemented algorithms turn out to be insecure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This is very related to the before-mentioned crypto-agility requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 4 Defining the la/[anagement Process In this section, we define our management process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In the context of this paper migration means replacing cryptographic primitives with those that can protect data from an attacker with access to a quantum computer powerful enough to break classic cryptography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To be able to acquire the resources, approval by the decision makers of the organization is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The migration to PQC is a complex task that requires substantial resources, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' more powerful hardware, time, or external support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In PMMP 7 the PQC Migration recommendation by ETSI, in is recommended to create the rote of a migration manager Who has to manage the migration [8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 15-16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In an organization that has an ISMS implemented, this Wright be the ISO, which is in Charge of managing the organization"s Information Security (See Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Organizations that are required to have an ISMS in place a d er are certified after ISO 27001 Standard, already have the needed Support fror management [7] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' There, Support is required as the management of Information Security happens fror top to bottom rather than fror bottom to top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To migrate successfully, the top management has to delegate tasks for the suceessful Integration of PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Figure l g i e s an overview of the PMMP steps and their interdependencíes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' It is highly recommended to execute the steps in their given oder, to assure a smooth flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For instanz, Fisk assessment relies on the inventory of cryptography as an Input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='Organization Management ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='Application Management ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='Executive Officer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='Risk Officer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='Security Officer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='Application Developer I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='System Administrator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='Educaflon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='Decide on time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='left ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='Define context of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='the organization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='› ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='Educaflon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='› Assess resources ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='› Provide Information ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='on Systems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='› Security policy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='- - - - - - - - - - - - - - - - - - - - - - - - - - | n f o r m a t i o n - - - - - - - - - - - - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='› Compile cryptographic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='inventory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='Assess risks < ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='Group and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='prioritize ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='applications ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='information- - - - - - - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='› ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' PMMP Overview 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='1 Education To get Support fror management, it is necessary to educate decision makers on the topic of post-quantum cryptography to ensure they know why there is a need to migrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Education Can take the Form of Workshops er Seminars where managers have the opportunity to ask questions and understand the dangers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' It is important to ensure the decision makers know what is Coming a d react in a seeure manne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' PMMP 7 the PQC migration recommendation by ETSI, it is recommended to create the role of a migration manager who has to manage the migration [8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 15-161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=" In an organization that has an ISMS implemented, this might be the ISO, which is in charge of managing the organization's information security (see Figure I)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Organizations that are required to have an ISMS in place and or are certified after IS() 27001 standard, already have the needed support from management [7] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' There, support is required as the management of information security happens from top to bottom rather than from bottom to top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To migrate successfully, the top management has to delegate tasks for the successful integration of PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Figure l gives an overview of the PMMP steps and their interdependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' It is highly recommended to execute the steps in their given order, to assure a smooth flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For instance, risk assessment relies on the inventory of cryptography as an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='Organization Management ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='Application Management ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='Executive Officer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='Risk Officer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='Security Officer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='Application Developer / ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='System Administrator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='Education ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='Decide on time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='left ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='Define context of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='the organization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='> ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='Education ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='> Assess resources ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='> Provide information ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='on systems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='> Security policy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='- - - - - - - - - - - - - - - - - - - - - - - - - - | n f o r m a t i o n - - - - - - - - - - - - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='> Compile cryptographic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='inventory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='Assess risks < ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='Group and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='prioritize ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='applications ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='lnformation- - - - - - - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='› ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' PMMP Overview 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='1 Education To get support from management, it is necessary to educate decision makers on the topic of post-quantum cryptography to ensure they know why there is a need to migrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Education can take the form of workshops or seminars where managers have the opportunity to ask questions and understand the dangers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' It is important to ensure the decision makers know what is coming and react in a secure manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 8 von Nethen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, the Software developers responsible for making changes in the applications hostess by the organization have to be educated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The programmes have to know how to komplement the needed algorithmus er how to correctly se cryptographic libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' If knowledge of how to develop Secure Software is not already established, the engineers have to acquire this knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Management has to provide unding to e fable this type of Security training (See also [9, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In the training is not sufficient er the organization does not have adequatere- sources for providing that Kind of education, the organization has to acquire external Support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For instanz, as the topic of PQC is currently under extensive research, the organization may involve universities er Institutes researching the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='2 Decide on time left Based on Mosca°s theorem [14], the Senior management estimates the account of time left before CRQCs are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This allows the development of a timeline on which further decisions Can be based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Of course, the decision has to be realistic and based upon estimates by authorities like the BSI er the NIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='3 Security poliey a d Goals The PQC Migration goals should be documented in the organization"s goals a d need so be published Inside The organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, the Security require- ments for reaching the goals have to be doeumented in an organization-wide Security policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Organizations applying an ISMS already have a Security policy that should be extended aceordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' New Software, either developed in-house er provided by third parties, an be required to Feature a certain level of cryptographic agility which an be used to (later) komplement PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To Set a Migration timeline, the policy has to deine the date on which CRQCs are thought to be a real thing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' As show in Figure l, the Security Officer is in Charge of developing the Security policy based on the management decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The policy an Feature concrete technical Statements Voith which algorithmus the organization warts to protect which data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For instanz, to exchange Keys Voith a party processing data requiring a medium level of Security, KYBER-768 Wright be Chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In addition, the Security policy has to document a list of excluded algorithmus if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This has to be coordinated Voith the Communication part- ners of the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Likewise, a list of preferred algorithmus Can be formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The decision is up to the organization because they know best which data has which proteetion requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Alternatively, the seeurity poliey can reference a technical document fror the BSI er the NIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='4 Context of the organization The context of the organization deines the scope in which cryptographic prim- itives need to be upgraded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Also, the scope deines where the organization has 8 von Nethen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, the software developers responsible for making changes in the applications hosted by the organization have to be educated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The programmers have to know how to implement the needed algorithms or how to correctly use cryptographic libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' If knowledge of how to develop secure software is not already established, the engineers have to acquire this knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Management has to provide funding to enable this type of security training (see also [9, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' $.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='l21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' If the training is not sufficient or the organization does not have adequatere- sources for providing that kind of education, the organization has to acquire external support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For instance, as the topic of PQC is currently under extensive research, the organization may involve universities or institutes researching the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content="2 Decide on time left Based on Mosca's theorem [141, the senior management estimates the amount of time left before CRQCs are available." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This allows the development of a timeline on which further decisions can be based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Of course, the decision has to be realistic and based upon estimates by authorities like the BSI or the NIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content="3 Security policy and goals The PQC migration goals should be documented in the organization's goals and need to be published inside the organization." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, the security require- ments for reaching the goals have to be documented in an organization-wide security policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Organizations applying an ISMS already have a security policy that should be extended accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' New software, either developed in-house or provided by third parties, can be required to feature a certain level of cryptographic agility which can be used to slater) implement PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To set a migration timeline, the policy has to define the date on which CRQCs are thought to be a real thing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' As shown in Figure I, the security officer is in charge of developing the security policy based on the management decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The policy can feature concrete technical statements with which algorithms the organization wants to protect which data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For instance, to exchange keys with a party processing data requiring a medium level of security, KYBER-768 might be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In addition, the security policy has to document a list of excluded algorithms if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This has to be coordinated with the communication part- ners of the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Likewise, a list of preferred algorithms can be formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The decision is up to the organization because they know best which data has which protection requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Alternatively, the security policy can reference a technical document from the BSI or the NIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='4 Context of the organization The context of the organization defines the scope in which cryptographic prim- itives need to be upgraded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Also, the scope defines where the organization has PMMP 9 to influence the upgrading of algorithmus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This Impacts the eomplexity of the Migration and the mount of work needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Below sections point out how the context of the organization Can be defined and hat it is Made of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In Figure 1, this step is situated in the column of the executive officer Who is in Charge of defining the context of the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Stakeholders Stakeholders have an Interest in the organization"s success und its conformity to laws, regulations, er contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Organizations have different Stakeholders: Customers, employees, partners, suppliers, regulatory authorities, a d themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' As the success of an organization an strongly depend on data seeurity, the Stakeholders have an Interest in Information Security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For instanz, an Interest in Information Security could be justified if the organization handles Customer data er private Information of its partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In effet, the organization an get obliged by its customers to migrate to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This Wright be the Case if a Service Level Agreement (SLA) exists between the organization and its eustomers, defining the level of Security, availability, er similar parameters defining the Service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The Other Way round: If the organization"s Service providers process data that is relevant to the Stakeholders, it must be examined to what extent the Service providers can be required by the organization to maintain the same level of Security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Managing the migration to PQC in the organization would in this Case also affect the suppliers of the organization, that Can get obliged to migrate to PQC As weil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' If the organization is a regulated organization, like a Financial Institute, it has to comply Voith certain l a s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Authorities Can require the organization to fulfill legal er regulatory requirements as it is the esse for Financial Institutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Protection of Customer data Can be required by data proteetion l a s such as the European General Data Protection Regulation (GDPR)3, which requires organizations processing personal data to Secure data u sing State of the alt techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Sorge day, State-of-the-art technology will include PQC and require organizations to adapt their cryptography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' As the Stakeholders also have an economie Interest in the success of the organization, they are required to agree upon the organization°s plans for migra- tion [16, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' l73-174].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This is because the migration requires financial resourees, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g, to provide upgraded hardware er to get external Support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' At last, Stakeholders can also be insurers Who Cover the organization against cyber risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' If standardized PQC algorithmus are not used, the Insurance may be invalidated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This Wright additionally be a driver of the migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Communication partners The Fisk quantum Computers pose apply especially to public-key algorithmus like RSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Public-key eryptography Systems are strongly used for exchanging symmetric Keys er to develop a shared key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The algoríthms are used to Secure data in transit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The data in transit is exchanged via insecure 3 See https://eur-lex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='europa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='eu/eli/reg/2016/679/oj PMMP 9 to influence the upgrading of algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This impacts the complexity of the migration and the amount of work needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Below sections point out how the context of the organization can be defined and what it is made of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In Figure 1, this step is situated in the column of the executive officer who is in charge of defining the context of the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=" Stakeholders Stakeholders have an interest in the organization's success and its conformity to laws, regulations, or contracts." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Qrganizations have different stakeholders: Customers, employees, partners, suppliers, regulatory authorities, and themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' As the success of an organization can strongly depend on data security, the stakeholders have an interest in information security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For instance, an interest in information security could be justified if the organization handles customer data or private information of its partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In effect, the organization can get obliged by its customers to migrate to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This might be the case if a Service Level Agreement (SLA) exists between the organization and its customers, defining the level of security, availability, or similar parameters defining the service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=" The other way around: If the organization's service providers process data that is relevant to the stakeholders, it must be examined to what extent the service providers can be required by the organization to maintain the same level of security." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Managing the migration to PQC in the organization would in this case also affect the suppliers of the organization, that can get obliged to migrate to PQC as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' If the organization is a regulated organization, like a financial institute, it has to comply with certain laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Authorities can require the organization to fulfill legal or regulatory requirements as it is the case for financial institutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Protection of customer data can be required by data protection laws such as the European General Data Protection Regulation (GDpR)3, which requires organizations processing personal data to secure data using state of the art techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Some day, state-of-the-art technology will include PQC and require organizations to adapt their cryptography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=" As the stakeholders also have an economic interest in the success of the organization, they are required to agree upon the organization's plans for migra- tion [16, pp." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 173-1741.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This is because the migration requires financial resources, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g, to provide upgraded hardware or to get external support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' At last, stakeholders can also be insurers who cover the organization against Cyber risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' If standardized PQC algorithms are not used, the insurance may be invalidated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This might additionally be a driver of the migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Communication partners The risk quantum computers pose apply especially to public-key algorithms like RSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Public-key cryptography systems are strongly used for exchanging symmetric keys or to develop a shared key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The algorithms are used to secure data in transit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The data in transit is exchanged via insecure 3 See https://eur-lex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='europa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='eu/eli/reg/2016/679/oj 10 von Nethen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Channels, like The Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Now, to Keep this data Secure, it is up to both side to encrypt und decrypt the data sing the same algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' So, a successful migration Can fly succeed if the Communication partners are taken long a d also migrate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' otherwise, there will be no Communication at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To prevent this fror happening, it is advised to not only compile an inventory of cryptography as explained in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='5, but also to compile an inventory of Communication partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The list of partners has to document hat their Communication endpoints Feature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', which cryptographic primitive(s) they are able to nun, in oder to exchange Information Voith the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, the organization a d its communíeation partners have to agree on an intersection of cryptographic primitives and parameters to ensure seeure Communication after migration to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Ideally, the algorithmus are supported on the hardware the partners already are u sing file fitting their needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For proper identification of the algorithmus, existing and standardized identifiers for algorithmus have to be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In the same manne, the partners should exclude algorithmus they do not wart to use (because of trust issues and alike).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' There are Communication partners Voith who the organization Carnot agree individually on algorithmus, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Users of a Website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Depending on the need for protection, these connections must be encrypted opportunistically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For this pur- pose, the usual Users of the Website must be analyzed in oder to determine which algorithmus Core i f o question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Then, they Can be provided Voith the best possible Security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' On a technieal layer, opportunistic Security is implemented weithin the applications that are migrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' With a so-called friends-and-family phase the migration of Selected appli- cations Can be tested a d evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' There, only a Small number of Selected Communication partners Upgrade their Systems to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' A successful prototyp- ical migration an her be used to advertise the use of PQC and get Other Communication partners to migrate also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Resources As noted in the previous section, the Management must provide the resources for the migration, er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Investments in the education of the employees, external Support, er Brewer hardware a d Software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Before money is spent, the organization should assess hat it Can Start without acquiring more resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' That is, estimating which Computing resourees are used to which extent a d how much room there is for algorithmus Voith higher resource demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Likewise, the resourees of the Software engineers need to be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Are they able to komplement new algorithmus er securely se Updates of cryptographic libraries featuring new cryptographic primitives?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In Figure l this relationship is resembled by the dotted arrow connecting "Assess resources" in the rightmost lane and the "Deine context of the organization" in the leftmost lane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='5 Inventory of cryptography To be üble to assess the Fisk a d the Impact of quantum Computing on the organization, the organization needs to compile an inventory of eryptography 10 von Nethen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' channels, like the internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Now, to keep this data secure, it is up to both sides to encrypt and decrypt the data using the same algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' So, a successful migration can only succeed if the communication partners are taken along and also migrate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' otherwise, there will be no communication at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To prevent this from happening, it is advised to not only compile an inventory of cryptography as explained in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='5, but also to compile an inventory of communication partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The list of partners has to document what their communication endpoints feature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', which cryptographic primitive(s) they are able to run, in order to exchange information with the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, the organization and its communication partners have to agree on an intersection of cryptographic primitives and parameters to ensure secure communication after migration to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Ideally, the algorithms are supported on the hardware the partners already are using while fitting their needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For proper identification of the algorithms, existing and standardized identifiers for algorithms have to be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In the same manner, the partners should exclude algorithms they do not want to use (because of trust issues and alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' There are communication partners with whom the organization cannot agree individually on algorithms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', users of a website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Depending on the need for protection, these connections must be encrypted opportunistically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For this pur- pose, the usual users of the website must be analyzed in order to determine which algorithms come into question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Then, they can be provided with the best possible security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' On a technical layer, opportunistic security is implemented within the applications that are migrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' With a so-called friends-and-family phase the migration of selected appli- cations can be tested and evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' There, only a small number of selected communication partners upgrade their systems to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' A successful prototyp- ical migration can then be used to advertise the use of PQC and get other communication partners to migrate also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Resources As noted in the previous section, the management must provide the resources for the migration, et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', investments in the education of the employees, external support, or newer hardware and software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Before money is spent, the organization should assess what it can start without acquiring more resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' That is, estimating which computing resources are used to which extent and how much room there is for algorithms with higher resource demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Likewise, the resources of the software engineers need to be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Are they able to implement new algorithms or securely use updates of cryptographic libraries featuring new cryptographic primitives?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In Figure 1 this relationship is resembled by the dotted arrow connecting "Assess resources" in the rightmost lane and the "Define context of the organization" in the leftmost lane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='5 Inventory of cryptography To be able to assess the risk and the impact of quantum computing on the organization, the organization needs to compile an inventory of cryptography PMMP 11 (also called crypto-inventory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Risk management Can the decide which Systems need to get migrated erst, based on their need for protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' White applying this method, all relevant Systems must be discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This ensures the migration process does not leave any Systems behind, undeteeted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The inventory provides a list of applications hat se cryptography and also Shows which algorithmus are used Voith which key lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This inventory serves as a starting point for migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In Figure 1, the creation of the crypto-inventory follows after defining the context of the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This is the task of Security officers, s i r e they has the resources to Interpret the technical details of the inventory and document the Security level of the applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The inventory an be compiled u sing various methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For instanz, if there is few Systems, the inventory an be compiled by hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Then, the administrators of a System an be asked about the used cryptographic primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For bigger environments, like a whole organization, Alnahawi et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' al prognose the development and usage of automated cryptography detection tools [1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 918].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' These tools could Scan Systems for, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', stored SSH Keys, libraries used in applications, er trusted Foot certificates of a PKI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Applications that are developed in-house by the organization and where it is possible to access the Source Code can be scanned Voith text-based tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For example, it Wright be sufficient to Scan the Source Code for the terms encrypt er decrypt to und the used cryptographic primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Furthermore, documents about formen Security audits and Interface descriptions of The Systems Can help eompile a list of relevant Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' It Wright also be possible to Scan the hole network traffic in an organization to detect cryptography used in the network, er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', TLS handshakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This Works be- cause TLS-protected applications (like HTTPS, LDAPS, FTPS, IMAPS, Open- VPN, e t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' ) se the TLS handshake to negotiate the eipher Suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Then, the Chosen algorithmus a d key lengths could be recorded [15, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 9-l0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Automated tools are also recommended by NIST in [2, ll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 220-221].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' However, there are still no tools hat could be recommended at the moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' PKIs need Special attention, as there are dependencies between Keys [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' A PKI can have many different entities and be rather complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' All entities in the PKI have to migrate for the structure to function, as the certificates issued Wright be installed on a large number of different devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' However, in a Way a PKI is already same Kind of a crypto-inventory itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' After the migration is done a d the applications are upgraded to PQC, the cryptographic inventory is Updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Then, the Fisk assessment can Start agar Voith the most recht State of the inventory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' While eompiling the crypto-inventory, it is important to doeument which data is protected by the used algorithmus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To understand the application"s data flow, reverse engineering of the System Wright be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This helps assessing the risks, as an be Seen in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='6 Assess risks The developed process a r s to be Fisk-driven, which is why the Fisk Assessment is on of The most important steps her migrating efficiently und effectívely PMMP 11 also called crypto-inventoryl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Risk management can then decide which systems need to get migrated first, based on their need for protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' While applying this method, all relevant systems must be discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This ensures the migration process does not leave any systems behind, undetected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The inventory provides a list of applications that use cryptography and also shows which algorithms are used with which key lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This inventory serves as a starting point for migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In Figure 1, the creation of the crypto-inventory follows after defining the context of the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This is the task of security officers, since they has the resources to interpret the technical details of the inventory and document the security level of the applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The inventory can be compiled using various methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For instance, if there is few systems, the inventory can be compiled by hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Then, the administrators of a system can be asked about the used cryptographic primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For bigger environments, like a whole organization, Alnahawi et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' al propose the development and usage of automated cryptography detection tools [1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 918].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' These tools could scan systems for, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', stored SSH keys, libraries used in applications, or trusted root certificates of a PKI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Applications that are developed in-house by the organization and where it is possible to access the source code can be scanned with text-based tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For example, it might be sufficient to scan the source code for the terms encrypt or decrypt to find the used cryptographic primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Furthermore, documents about former security audits and interface descriptions of the systems can help compile a list of relevant systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' It might also be possible to scan the whole network traffic in an organization to detect cryptography used in the network, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', TLS handshakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This works be- cause TELS-protected applications (like HTTPS, LDAPS, FTPS, IMAPS, Open- VPN, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=') use the TLS handshake to negotiate the cipher suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Then, the chosen algorithms and key lengths could be recorded [15, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 9-I0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Automated tools are also recommended by NIST in [2, II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 220-221].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' However, there are still no tools that could be recommended at the moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' PKIs need special attention, as there are dependencies between keys [81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' A PKI can have many different entities and be rather complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' All entities in the PKI have to migrate for the structure to function, as the certificates issued might be installed on a large number of different devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' However, in a way a PKI is already some kind of a crypto-inventory itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' After the migration is done and the applications are upgraded to PQC, the cryptographic inventory is updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Then, the risk assessment can start again with the most recent state of the inventory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' While compiling the crypto-inventory, it is important to document which data is protected by the used algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=" To understand the application's data flow, reverse engineering of the system might be required." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This helps assessing the risks, as can be seen in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='6 Assess risks The developed process aims to be risk-driven, which is why the risk assessment is one of the most important steps when migrating efficiently and effectively 12 von Nethen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' CRQCs Can pose a Fisk to the data Security of an organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' As part of Fisk management, the Migration to PQC is considered a Fisk-minimizing measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This section explains the Interactions between Fisk management a d the migration process and Shows how different risks should be handled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In Figure 1, this step is situated in the lane of the Fisk ofñcer, which is responsible for the Fisk assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The application developers ( a d maintainers) are responsible for providing Information on the Systems that are subject to the Fisk assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Therefore, there is an arrow connecting the toto rotes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Supporting documents For each business process that is backed by different applications the risks CRQCs pose need to be assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' A business impaet anal- ysis Can be helpful at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The Impact analysis Shows how the business is affected in Case of interruptions to the Services provided by the IT Infrastructure of the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Moreover, the Impact analysis helps to justify expenses for the Migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, existing documents on business continuation Can help to assess the risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' With the additional help of a erypto-inventory, the ap- plications at Fisk an be identified, s i r e the inventory Shows which applications are sing algorithmus endangered by quantum Computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Assessing the Fisk in- volves evaluating how lang the data protected by the algorithmus used in the application needs to stay seeure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' If the time agreed upon by whieh CRQCs are available is shorter han the time that is left to migrate,4 the Fisk is high and the studied System has to be one of the first to get migrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Scope White applying the aforementioned techniques, that is, sing the resultat of the business Impact analysis a d the inventory of cryptography, the Fisk as- sessment has to over the hole scope of the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This includes the Stakeholders of the organization, which Wright be business partners, the organi- zation itself, er authorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In the previous sections, it b e e r e Clear that organizations have different Stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Not migrating to PQC Can have consequences that Stakeholders Wright er Wright not wart to accept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For instanz, a Customer that fels their data is no langer adequately protected could terminator the contract Voith the organization er Ile a lawsuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Authorities Wright impose severe penalties on the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Furthermore, Communication partners that do not (wart to) mi- grate to u sing PQC, Wright Fisk getting excluded fror the business.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Possible risks An attaeker Voith Access to a quantum Computer Wright deprive the organization of its business.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' It is not possible to migrate all Systems at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For this reason, in same Cases, the risks must be accepted at least temporarily before migrating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 4 Following The lines of hat is knoten As Mosca"s theorem, the time that is left to migrate to PQC is limited by the time needed to develop, prepare und deploy the new scheue plus the time needed for secrets to refrain Secure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 12 von Nethen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' CRQCs can pose a risk to the data security of an organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' As part of risk management, the migration to PQC is considered a risk-minimizing measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This section explains the interactions between risk management and the migration process and shows how different risks should be handled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In Figure 1, this step is situated in the lane of the risk officer, which is responsible for the risk assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The application developers (and maintainers are responsible for providing information on the systems that are subject to the risk assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Therefore, there is an arrow connecting the two roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Supporting documents For each business process that is backed by different applications the risks CRQCs pose need to be assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' A business impact anal- ysis can be helpful at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The impact analysis shows how the business is affected in case of interruptions to the services provided by the IT infrastructure of the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Moreover, the impact analysis helps to justify expenses for the migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, existing documents on business continuation can help to assess the risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' With the additional help of a crypto-inventory, the ap- plications at risk can be identified, since the inventory shows which applications are using algorithms endangered by quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Assessing the risk in- volves evaluating how long the data protected by the algorithms used in the application needs to stay secure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' If the time agreed upon by which CRQCs are available is shorter than the time that is left to migrate,4 the risk is high and the studied system has to be one of the first to get migrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Scope While applying the aforementioned techniques, that is, using the results of the business impact analysis and the inventory of cryptography, the risk as- sessment has to cover the whole scope of the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This includes the stakeholders of the organization, which might he business partners, the organi- zation itself, or authorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In the previous sections, it became clear that organizations have different stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Not migrating to PQC can have consequences that stakeholders might or might not want to accept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For instance, a customer that feels their data is no longer adequately protected could terminate the contract with the organization or file a lawsuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Authorities might impose severe penalties on the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Furthermore, communication partners that do not (want to) mi- grate to using PQC, might risk getting excluded from the business.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Possible risks An attacker with access to a quantum computer might deprive the organization of its business.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' It is not possible to migrate all systems at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For this reason, in some cases, the risks must be accepted at least temporarily before migrating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=" 4 Following the lines of what is known as Mosca's theorem, the time that is left to migrate to PQC is limited by the time needed to develop, prepare and deploy the new scheme plus the time needed for secrets to remain secure." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' PMMP 13 But, thinking of The performance requirements of the new algorithmus, there is Brother Fisk: Sorge algorithmus providing a high level of Security Can Slow down the Initial Connection to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', a Website for up to many seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The organization either has to accept the Fisk of losing eustomers beeause their Website appears to be very Slow, er Upgrade the hardware at least on their Side of the Connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Also, it an turn out that it is too hard to migrate an applieation so that a replacement would be better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The post-quantum algorithmus may turn out to be insecure during the migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' A required refurbishment er reverse-engineering of an applieation Wright prove to be too eostly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The organization°s Fisk management provides the deeision of whieh Systems to migrate and when.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Note that this deeision is eompletely Fisk-driven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For instanz, Systems that operator in the inne network of the organization Wright not get migrated erst as the Fisk is not high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Although they could be migrated Voith a simple Upgrade in less than a few minutes, they handle unimportant data a d are not reachable Over the Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='7 Risk- and process-based grouping a d prioritization of applications Organizations use different applications to operator their business.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The inter- operability of the Systems must be ensured, s i r e any error may Interrupt the organization7s processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To prevent such issues, the applications and Systems are grouped by the business process they are tied to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Then, the System Groups are migrated on by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In Figure l, the grouping of the applications is placed in the lane of the Fisk officer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Preceding is the Fisk assessment of the Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To recall the mitigation of the Y2K-Bug: It is imposant to remember that the organization only has limited resources of Software engineers, which is why Putnam and Schultz prognose to triage the Systems needing a migration [17, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 96] [18, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 65-66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' First, the Systems that involve life a d death need to be touched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Second, "if you are not in a life-and-death business" [17, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 96], the Systems Critical for running the organization°s business need to be worked on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The Systems to be handled last are those hose failure would be irritating, but not costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' These Systems could be handled after l January 2000 (YZK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In oder to adopt the process presented in [17] to PQC migration, the Systems need to get triaged by their criticality regarding the data they process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To conclude, the decision on which group is migrated erst is based on the Fisk assessment done before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='8 Testing a d monitoring The Migration to PQC in an organization is controlled by the management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' It is a Fisk-driven process that is Integrated into the ISMS of an organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Software mígrations Can take quite a lang time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Therefore, the processes defined in PMMP need to be monitored for effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Being a Cross layer aspect, the testing a d monitoring step is not displayed In Figure l for better readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' It PMMP 13 But, thinking of the performance requirements of the new algorithms, there is another risk: Some algorithms providing a high level of security can slow down the initial connection to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', a website for up to many seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The organization either has to accept the risk of losing customers because their website appears to be very slow, or upgrade the hardware at least on their side of the connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Also, it can turn out that it is too hard to migrate an application so that a replacement would be better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The post-quantum algorithms may turn out to be insecure during the migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' A required refurbishment or reverse-engineering of an application might prove to be too costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=" The organization's risk management provides the decision of which systems to migrate and when." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Note that this decision is completely risk-driven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For instance, systems that operate in the inner network of the organization might not get migrated first as the risk is not high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Although they could be migrated with a simple upgrade in less than a few minutes, they handle unimportant data and are not reachable over the internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='7 Risk- and process-based grouping and prioritization of applications Organizations use different applications to operate their business.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=" The inter- operability of the systems must be ensured, since any error may interrupt the organization's processes." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To prevent such issues, the applications and systems are grouped by the business process they are tied to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Then, the system groups are migrated one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In Figure 1, the grouping of the applications is placed in the lane of the risk officer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Preceding is the risk assessment of the systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To recall the mitigation of the YZK-Bug: It is important to remember that the organization only has limited resources of software engineers, which is why Putnam and Schultz propose to triage the systems needing a migration [17, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 961 [18, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 65-661.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' First, the systems that involve life and death need to be touched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Second, "if you are not in a life-and-death business" [17, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=" 961, the systems critical for running the organization's business need to be worked on." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The systems to be handled last are those whose failure would be irritating, but not costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' These systems could be handled after 1 January 2000 (YQK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In order to adopt the process presented in [17] to PQC migration, the systems need to get triaged by their criticality regarding the data they process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To conclude, the decision on which group is migrated first is based on the risk assessment done before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='8 Testing and monitoring The migration to PQC in an organization is controlled by the management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' It is a risk-driven process that is integrated into the ISMS of an organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Software migrations can take quite a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Therefore, the processes defined in PMMP need to be monitored for effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Being a cross layer aspect, the testing and monitoring step is not displayed In Figure l for better readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' It 14 von Nethen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' is the Task of the Security officer to enforee the previously defined Security policy (by ensuring the Migration processes do not top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For instanz, regulated organizations like Financial Institutes that already have an Internal control System in place ean use it to Monitor the effeetiveness of the developed migration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For example, a monitoring process Wright Check if the migration processes defined an be applied correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The monitoring has to detect a lack of resources required for migrating, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', lacking knowledge in the Geld of PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In such a Case, the management has to improve resources by organizing Workshops, establishing cooperation Voith universities, er releasing more financial resources for the education of the developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To ensure that the methods a d techniques of PQC are applied correctly, Security audits performed by external organizations an help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Also, Internal se- curity audits that the Security officer performs are helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 5 Evaluation In This Chapter, we verify that PMMP m e t s the defined requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Since PMMP is based on the presence of a reestablished ISMS und its respective Change processes the application of PMMP is possible wherever such a management System is in place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, PMMP is compared to the migration approaches presented in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='1 Meeting the requirements Migration Timeline PMMP Starts Voith educating Senior Management, to ensure the topic is understood by the decision makers in the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Then, the executive need to commit to a date by which CRQCs will be available a d theater the currently used cryptography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Based on the Fisk assessment and Voith the help of the inventory of cryptography, a migration timeline Can be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To allow exeeutives to estimate the duration of the migration steps, PMMP helps approximativ the mount of work needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' That is, the resources an orga- nization has are estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Which application is getting migrated First is a risk- based decision, not infiuencing the resources needed for the migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' U sing techniques like reverse engineering to understand an application"s architecture, the mount of work needed for the migration an be assessed (for example by measuring the complexity of the application sing the number of lines of Code) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Also, the migration process allows falling back to replacing the legacy applica- tion Voith a new o n , if it is fester (er easier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To develop a migration timeline, the process requires the User to Set the relevant dates stemming fror Mosca"s theorem: The time when a CRQC will be available, the duration in which data has to stay Secure, a d the duration of the migration to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Security PMMP Features the education of not fly the decision makers but also the education of, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', developers that komplement the new primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This provides solid ground on which the algorithmus an be applied correetly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 14 von Nethen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' is the task of the security officer to enforce the previously defined security policy (by ensuring the migration processes do not stop).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For instance, regulated organizations like financial institutes that already have an internal control system in place can use it to monitor the effectiveness of the developed migration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For example, a monitoring process might check if the migration processes defined can be applied correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The monitoring has to detect a lack of resources required for migrating, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', lacking knowledge in the field of PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In such a case, the management has to improve resources by organizing workshops, establishing cooperation with universities, or releasing more financial resources for the education of the developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To ensure that the methods and techniques of PQC are applied correctly, security audits performed by external organizations can help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Also, internal se- curity audits that the security officer performs are helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 5 Evaluation In this chapter, we verify that PMMP meets the defined requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Since PMMP is based on the presence of a reestablished ISMS and its respective change processes the application of PMMP is possible wherever such a management system is in place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, PMMP is compared to the migration approaches presented in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='1 Meeting the requirements Migration Timeline PMMP starts with educating senior management, to ensure the topic is understood by the decision makers in the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Then, the executives need to commit to a date by which CRQCs will be available and threaten the currently used cryptography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Based on the risk assessment and with the help of the inventory of cryptography, a migration timeline can be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To allow executives to estimate the duration of the migration steps, PMMP helps approximate the amount of work needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' That is, the resources an orga- nization has are estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Which application is getting migrated first is a risk- based decision, not influencing the resources needed for the migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=" Using techniques like reverse engineering to understand an application's architecture, the amount of work needed for the migration can be assessed (for example by measuring the complexity of the application using the number of lines of code) ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Also, the migration process allows falling back to replacing the legacy applica- tion with a new one, if it is faster (or easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=" To develop a migration timeline, the process requires the user to set the relevant dates stemming from Mosca's theorem: The time when a CRQC will be available, the duration in which data has to stay secure, and the duration of the migration to PQC." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Security PMMP features the education of not only the decision makers but also the education of, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', developers that implement the new primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This provides solid ground on which the algorithms can be applied correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' PMMP 15 Additionally, it is required that the process handles advancements in cryp- tographic Analysis, which Gould weaken the Most promising PQC algorithmus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' PMMP solves this by integrating it into the Fisk Management of the organi- zation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' As s o n as one cryptographie primitive is weakened, the Situation is evaluated agar, s i r e the Fisk management and the Fisk assessment are not a one-shot process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Then, the migration process has to Start again Voith adapted Security policies and algorithm selections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Completeness PMMP makes se of the Features of an established ISMS, such as System a d structure analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This is complemented by the creation of an inventory of cryptography that serves, amor Other purposes, the compilation of a list of migration-relevant Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Note that the formen a d the latte fackle a Common task fror different angles, thereby complementing each Other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Context awareness The migration process is required to be context-aware, As the Migration to PQC must be coordinated Voith business partners, to ensure the partners Can communicate Voith each Other file migrating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' PMMP solves this by involving business partners a d customers early in the process, ensuring the relevant Communication partners have the same Vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, the process takes into account the needs of Stakeholders a d those of regulatory authorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To ensure Communication partners of the organization are not exeluded file migrating, PMMP features a process to detect Communication partners, which includes the primitives each partner Supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The selection of suitable algorithmus for different applieations is enabled by complying Voith respective recommendations such as published by NIST er BSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Crypto-agility PMMP enables the establishment of erypto-agility by support- ing the fulfillment of the requirements for practiced crypto-agility As defined in the cryptographic agility maturity Model (CAMM) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' A More detailed pre- sentation of the respective CAMM requirements ad how they are supported by PMMP is g i e n in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Interoperability To prevent interrupting business processes, interoperabilíty between the Systems is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' PMMP advises to se gateways between ex- isting Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Then, Systems that are not yet upgraded to PQC Can connect sing the gateway technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Also, Systems that are not able to be migrated, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', because the Software vendors Carnot er do not wart to komplement PQC, er because the Source Code of the Software Carnot be modified, can make use of gateways and ensure the interoperabílíty of the Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, the migration process is based on a process- and Fisk-based groupíng of the applications a d Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Thereby, the interoperabílíty of the Systems an be ensured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' PMMP 15 Additionally, it is required that the process handles advancements in cryp- tographic analysis, which could weaken the most promising PQC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' PMMP solves this by integrating it into the risk management of the organi- zation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' As soon as one cryptographic primitive is weakened, the situation is evaluated again, since the risk management and the risk assessment are not a one-shot process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Then, the migration process has to start again with adapted security policies and algorithm selections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Completeness PMMP makes use of the features of an established ISMS, such as system and structure analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This is complemented by the creation of an inventory of cryptography that serves, among other purposes, the compilation of a list of migration-relevant systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Note that the former and the latter tackle a common task from different angles, thereby complementing each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Context awareness The migration process is required to be context-aware, as the migration to PQC must be coordinated with business partners, to ensure the partners can communicate with each other while migrating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' PMMP solves this by involving business partners and customers early in the process, ensuring the relevant communication partners have the same vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, the process takes into account the needs of stakeholders and those of regulatory authorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To ensure communication partners of the organization are not excluded while migrating, PMMP features a process to detect communication partners, which includes the primitives each partner supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The selection of suitable algorithms for different applications is enabled by complying with respective recommendations such as published by NIST or BSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Crypto-agility PMMP enables the establishment of crypto-agility by support- ing the fulfillment of the requirements for practiced crypto-agility as defined in the cryptographic agility maturity model (CAMM) 1101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' A more detailed pre- sentation of the respective CAMM requirements ad how they are supported by PMMP is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Interoperability To prevent interrupting business processes, interoperability between the systems is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' PMMP advises to use gateways between ex- isting systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Then, systems that are not yet upgraded to PQC can connect using the gateway technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Also, systems that are not able to be migrated, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', because the software vendors cannot or do not want to implement PQC, or because the source code of the software cannot be modified, can make use of gateways and ensure the interoperability of the systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, the migration process is based on a process- and risk-based grouping of the applications and systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Thereby, the interoperability of the systems can be ensured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 16 von Nethen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Interim resultat Because the migration to PQC Can take a lang time, the Migration process is required to deine a d deliver Interim resultat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Interim resultat are delivered in various steps of the Migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The erst step of the Migration is to educate the executive management on the topic of quantum Computing and its Fisk to classic cryptography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The knowledge gained in this step is an Interim result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Later in the migration, her the migration strategy is formed, there exit dif- ferent approaches for global migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' When the organization decides to takle the migration sing the incremental packet conversion approach, this method delivers numerous Interim resultat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The application parts that are incrementally put into produetion resemble Interim resultat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' By sing gateway technologies it can be ensured that the applications stay compatible Voith the rest of the orga- nization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Also, by rating awareness for the topic of PQC, a snowball effet an be triggered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' When Stakeholders of the organization are required to migrate to PQC, this triggers stakeholder7s Stakeholders to migrate also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This is also an intermediate result in the global migration to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Rea ct to advancements in cryptanalysis The Fisk Assessment of the organi- zation is not a one-shot process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The Fisk is assessed regularly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Therefore, her advancements in cryptanalysis theater PQC er the knoten classic cryptographic primitives, the migration process has to be applied agar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' When advancements are Made file migrating, the Migration is adapted aceordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='2 Comparison to existing migration approaches One of the Most nature approaches existing is the on developed by Zhang et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' al, presented in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' What the approaches have in Common, is that the Migration to PQC is triggered fror the top of the organization, initiated by the decision makers that get edueated on the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The Second step is to eompile an inventory of cryptography, but it is not mentioned how the inventory can be compiled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Regarding the migration of on Single application (the IBM Db2 database), the compilation of the inventory is not in their Focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Moreover, the toto approaches focus on the management of the migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In [22], step Six requires the User of the approach to execute the cybersecurity policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The policy Controls the selection of appropriate Solutions based on the requirements of the organization and budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' However, it refrains unclear how the requirements can be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' While the approaeh by Zhang et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' al suggestiv Working on the Systems that handle Critical data erst, it is not explained, what Critical data is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Here, a Fisk-based process would be better suited, as proposed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The migration approach presented in [12] Features a Fisk-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' When the quantum Fisk for an organization is high, the approaeh suggestiv imple- menting hybrid cryptography as sonn as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' If the quantum Fisk is low, the approach proposes to wart for an update for the application(s) in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The problem Voith this approach is that the Fisk assessment is done before eompiling 16 von Nethen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Interim results Because the migration to PQC can take a long time, the migration process is required to define and deliver interim results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Interim results are delivered in various steps of the migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The first step of the migration is to educate the executive management on the topic of quantum computing and its risk to classic cryptography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The knowledge gained in this step is an interim result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Later in the migration, when the migration strategy is formed, there exist dif- ferent approaches for global migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' When the organization decides to tackle the migration using the incremental packet conversion approach, this method delivers numerous interim results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The application parts that are incrementally put into production resemble interim results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' By using gateway technologies it can be ensured that the applications stay compatible with the rest of the orga- nization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Also, by raising awareness for the topic of PQC, a snowball effect can be triggered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=" When stakeholders of the organization are required to migrate to PQC, this triggers stakeholder's stakeholders to migrate also." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This is also an intermediate result in the global migration to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' React to advancements in cryptanalysis The risk assessment of the organi- zation is not a one-shot process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The risk is assessed regularly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Therefore, when advancements in cryptanalysis threaten PQC or the known classic cryptographic primitives, the migration process has to be applied again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' When advancements are made while migrating, the migration is adapted accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='2 Comparison to existing migration approaches One of the most mature approaches existing is the one developed by Zhang et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' al, presented in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' What the approaches have in common, is that the migration to PQC is triggered from the top of the organization, initiated by the decision makers that get educated on the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The second step is to compile an inventory of cryptography, but it is not mentioned how the inventory can be compiled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Regarding the migration of one single application (the IBM Db2 databased, the compilation of the inventory is not in their focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Moreover, the two approaches focus on the management of the migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In [221, step six requires the user of the approach to execute the cybersecurity policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The policy controls the selection of appropriate solutions based on the requirements of the organization and budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' However, it remains unclear how the requirements can be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' While the approach by Zhang et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' al suggests working on the systems that handle critical data First, it is not explained, what critical data is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Here, a risk-based process would be better suited, as proposed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The migration approach presented in [12] features a risk-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' When the quantum risk for an organization is high, the approach suggests imple- menting hybrid cryptography as soon as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' If the quantum risk is low, the approach proposes to wait for an update for the application(s) in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The problem with this approach is that the risk assessment is done before compiling PMMP 17 the crypto-inventory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' It is unclear how The Fisk Can be assessed if it is unknown which applications se which type of cryptography, which key length und for which data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' PMMP puts Fisk management in Focus a d takes the advances in quantum Computing as a Fisk that needs to be mitigated by the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The basis for this process is the compilation of a crypto-inventory before as- sessing the Fisk, s i r e the inventory is needed for the Fisk assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' What the approaches have in Common is that the approach presented in [12] also seeks to get Support fror the decision makers in the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' One key differenz is that PMMP integrales into existing management pro- cesses provided, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', by a nature Information Management Security System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' It uses the steering mechanismus in an organization for the migration to PQC fror top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' While applying the steps of PMMP Voith the testing and monitoring in place, features of the Internal control System are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 6 Aehievements a d open issues In the previous sections, PMMP a process for Managing the Migration to PQC is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To develop a process hat is oriented to the needs of the industrie, requirements were Set up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Then, the process was evaluated against these re- quirements, to See how the migration process performs (in comparison to Other approaches) ø The successful evaluation of PMMP Shows that it can be used for migrating fror classic cryptography to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' It is explained, which challenges have to be solved a d how existing management methods an be used in the context of migrating to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To achieve that, PMMP uses a Fisk-based method, regarding the advances in quantum Computing as a Fisk that needs to be handled by Fisk management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Also, the process includes Interim States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' When u sing the incremental packet conversion approach, every application that is migrated to PQC and put into production resembles an Interim State.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Furthermore, the process puts a strong emphasis on the context of the organization including its Stakeholders and com- munication partners, Voith the latte berg one of the most important aspects of a successful migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This process Supports the cryptographie Understanding of organizations: To be able to migrate to PQC, the organizations have to un- derstand which applications are vulnerable a d what level of Security hey need to provide for the data processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Although a migration process was developed, not every application Can be migrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' There still exit applications that Carnot get migrated, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', due to license reasons er because the Source ende is not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This problem an be solved sing a technique presented by Sneed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' [9, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 4], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' sing gateways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For instanz, a PQC-VPN can be used to ensure applications communicate sing PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' PMMP 17 the crypto-inventory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' It is unclear how the risk can be assessed if it is unknown which applications use which type of cryptography, which key length and for which data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' PMMP puts risk management in focus and takes the advances in quantum computing as a risk that needs to be mitigated by the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The basis for this process is the compilation of a crypto-inventory before as- sessing the risk, since the inventory is needed for the risk assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' What the approaches have in common is that the approach presented in [12] also seeks to get support from the decision makers in the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' One key difference is that PMMP integrates into existing management pro- cesses provided, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', by a mature Information Management Security System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' It uses the steering mechanisms in an organization for the migration to PQC from top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' While applying the steps of PMMP with the testing and monitoring in place, features of the internal control system are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 6 Achievements and open issues In the previous sections, PMMP a process for managing the migration to PQC is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To develop a process that is oriented to the needs of the industry, requirements were set up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Then, the process was evaluated against these re- quirements, to see how the migration process performs (in comparison to other approaches) • The successful evaluation of PMMP shows that it can be used for migrating from classic cryptography to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' It is explained, which challenges have to he solved and how existing management methods can be used in the context of migrating to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To achieve that, PMMP uses a risk-based method, regarding the advances in quantum computing as a risk that needs to be handled by risk management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Also, the process includes interim states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' When using the incremental packet conversion approach, every application that is migrated to PQC and put into production resembles an interim state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Furthermore, the process puts a strong emphasis on the context of the organization including its stakeholders and com- munication partners, with the latter being one of the most important aspects of a successful migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This process supports the cryptographic understanding of organizations: To he able to migrate to PQC, the organizations have to un- derstand which applications are vulnerable and what level of security they need to provide for the data processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Although a migration process was developed, not every application can be migrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' There still exist applications that cannot get migrated, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', due to license reasons or because the source code is not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This problem can be solved using a technique presented by Sneed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' [9, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 4], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' using gateways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For instance, a PQC-VPN can be used to ensure applications communicate using PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 18 von Nethen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='1 Open i es SS Not all issues Gould be solved file developing the Migration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' There refrain a few open issues: Earlier, it is mentioned hat the organization has to assess the resources it has to migrate to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' There e i s t resources that can easily be quantified, such as the Financial backing er the amputation power of the Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' But, also developer resources need to be aecessed in oder to evaluate the need to provide external Support (fror Other organizations er universities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' If there is a metric, that can be used to assess the eompetenee of Software engineers, it should be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' One of the most important tasks when migrating is to compile the inventory of cryptography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' When applying the developed migration process it may oeeur that Systems are left out er overlooked a d do not get migrated to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' These Systems pose a Security vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To the best of nur knowledge, to this day, there is no conerete tool that can be recommended, as the researeh is still ongoing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Nevertheless, due to the Fisk-based assessment of the Systems and the cryptographic inventory, forgetting Systems is unlikely to happen Voith PMMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Further, if one System is left out in the migration, Other Systems may not be able to connect to this System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Then, the overlooked System automatically receives the attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In analogy to the development life Cycle of maturity models presented by Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' [3], the development of a migration process is an iterative process that includes feedback fror outside experts a d practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In the work at hand we present nur erst proposal for PMMP to obtain external feedbaek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' A real-world migration has not yet been executed a d it refrains open how PMMP will perform her applied in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 7 Conclusion In This paper, the Migration to PQC was discussed fror different points of View.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' A Management process for the migration was developed that is oriented to the needs of the industrie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Throughout the paper, it was argued hat organizations that wart to migrate to PQC have to Start as early as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The possible threats fror advances in the Geld of quantum Computing Carnot be unseen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' With the developed migration process, a concept hat helps fackle this Challenge was formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' An open question is how the open-souree Community will react to further advances in quantum Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' If IBM keeps its promises, powerful quantum Computers will be cracking RSA sooner than later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Maybe a Worldwide Wave of Innovation will bring PQC to many open-source products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Cryptography is broadly used for securing traffic on the Internet, Voith HTTPS berg one major use esse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' As Google a d Cloudfiare showed, the devices of most Internet Users are capable of running post-quantum Secure algorithmus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' If Companies (like the latte toto) use their large market power to initialize the migration to PQC, so 18 von Nethen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='1 Open i u s SS Not all issues could be solved while developing the migration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' There remain a few open issues: Earlier, it is mentioned that the organization has to assess the resources it has to migrate to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' There exist resources that can easily be quantified, such as the financial backing or the computation power of the systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' But, also developer resources need to be accessed in order to evaluate the need to provide external support (from other organizations or universities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' If there is a metric, that can be used to assess the competence of software engineers, it should be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' One of the most important tasks when migrating is to compile the inventory of cryptography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' When applying the developed migration process it may occur that systems are left out or overlooked and do not get migrated to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' These systems pose a security vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To the best of our knowledge, to this day, there is no concrete tool that can be recommended, as the research is still ongoing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Nevertheless, due to the risk-based assessment of the systems and the cryptographic inventory, forgetting systems is unlikely to happen with PMMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Further, if one system is left out in the migration, other systems may not be able to connect to this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Then, the overlooked system automatically receives the attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In analogy to the development life cycle of maturity models presented by Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' [3], the development of a migration process is an iterative process that includes feedback from outside experts and practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In the work at hand we present our first proposal for PMMP to obtain external feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' A real-world migration has not yet been executed and it remains open how PMMP will perform when applied in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 7 Conclusion In this paper, the migration to PQC was discussed from different points of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' A management process for the migration was developed that is oriented to the needs of the industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Throughout the paper, it was argued that organizations that want to migrate to PQC have to start as early as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The possible threats from advances in the field of quantum computing cannot be unseen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' With the developed migration process, a concept that helps tackle this challenge was formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' An open question is how the open-source community will react to further advances in quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' If IBM keeps its promises, powerful quantum computers will be eraeking RSA sooner than later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Maybe a worldwide wave of innovation will bring PQC to many open-source products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Cryptography is broadly used for securing traffic on the internet, with HTTPS being one major use ease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' As Google and Cloudflare showed, the devices of most internet users are capable of running post-quantum secure algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' If companies (like the latter two use their large market power to initialize the migration to PQC, so PMMP 19 that smaller organizations Wright feel the need to migrate, the global migration Gould be sped up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Also, manufacturers of web browsers have strong power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For example, the popular web browser Firefox displays a war fing ("connector not seeure") when Users connect to an HTTP-only Site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The current Version of Firefox also prevents connecting to a web Server sing the deprecated TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='0 er 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='1 a d displays a war fing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For an organization, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', a newspaper er an online Shop, this Can result in a bad reputation, because the Users See an error when connecting to the page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' If one day quantum Computers are powerful enough, we may See a similar war fing in web browsers we use then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Most promising Wright be an update of widely used frameworks like OpenSSL to feature PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' A new Version of the framework that implements the new algorithmus Gould bring PQC to many devices, as lang as they are powerful enough to nun the new primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To get the migration on the Way, it is needed to repeat the very First step of the migration process presented in this paper: Educate executives and decision makers in PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Only if the possible dangers to the cryptography used today are recognized, someone will Take the Money und Change Something.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Recalling the similarity of the Migration fror IPv4 to IPv6: there are net- works that have already completely switched to sing the Brewer protocol, those that se both variant, and those that still only se the oder variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This devel- opment will very likely also be Seen in the migration to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In the beginning, there will be only a few networks migrated to use post-quantum key exchanges er encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In the meantime, there will be networks supporting both (er all three) possible variants: classic cryptography, hybrid cryptography, a d PQC only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Also, NIST will most likely publish the Standards specifying PQC in the next toto year (fror 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' With these Standards, organizations can decide more easily which algorithmus they need to komplement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Further work on the topic Wright include u sing the developed migration process, improving it, a d helping organizations migrate to PQC by increasing their cryptographic agility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Meanwhile, we fall See, how the global migration to PQC will move Forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' References l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Alnahawi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Wiesmaier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Grasmeyer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Geißler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Zeier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Bauspieíš, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Heinemann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=': On the State of post-quantum cryptography migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Jahrestagung der Gesellschaft für Informatik, INFORMATIK 2021 - Computer Science & Sustainability, Berlin, Germany, 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' September - l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Oktober, 2021 P- 314, 907-941 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='18420/infolmatik2021-078, https : //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='18420/informatik2021-078 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Barker, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Polk, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Souppaya, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=': Getting Ready for Post-Quantum Cryp- tography:: Explore Challenges Associated Voith Adoption a d Use of Post- Quantum Cryptographic Algorithmus (May 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='6028/ NIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='CSWP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='05262020-draft, https://nvlpubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='gov/nistpubs/CSWP/NIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' CSWP • 05262020-draf t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='pdf 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Becker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Knackstedt, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Pöppelbulš, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=': Developing maturity models for management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' I n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' (3), 213-222 (06 2009) 1 IT PMMP 19 that smaller organizations might feel the need to migrate, the global migration could be sped up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Also, manufacturers of web browsers have strong power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For example, the popular web browser Firefox displays a warning ("connection not secure") when users connect to an HTTP-only site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The current version of Firefox also prevents connecting to a web server using the deprecated TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='1 and displays a warning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For an organization, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', a newspaper or an online shop, this can result in a bad reputation, because the users see an error when connecting to the page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' If one day quantum computers are powerful enough, we may see a similar warning in web browsers we use then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Most promising might be an update of widely used frameworks like OpenSSL to feature PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' A new version of the framework that implements the new algorithms could bring PQC to many devices, as long as they are powerful enough to run the new primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To get the migration on the way, it is needed to repeat the very first step of the migration process presented in this paper: Educate executives and decision makers in PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Only if the possible dangers to the cryptography used today are recognized, someone will take the money and change something.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Recalling the similarity of the migration from IPv4 to IPv6: there are net- works that have already completely switched to using the newer protocol, those that use both variants, and those that still only use the older variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This devel- opment will very likely also be seen in the migration to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In the beginning, there will be only a few networks migrated to use post-quantum key exchanges or encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In the meantime, there will be networks supporting both for all three) possible variants: classic cryptography, hybrid cryptography, and PQC only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Also, NIST will most likely publish the standards specifying PQC in the next two years (from 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' With these standards, organizations can decide more easily which algorithms they need to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Further work on the topic might include using the developed migration process, improving it, and helping organizations migrate to PQC by increasing their cryptographic agility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Meanwhile, we shall see, how the global migration to PQC will move forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' References l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Alnahawi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Wiesmaier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Grasmeyer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Geiiéler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Zeier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Bauspielé, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Heinemann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=': On the state of post-quantum cryptography migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Jahrestagung der Gesellschaft fur Information, INFCRMATIK 2021 - Computer Science & Sustainability, Berlin, Germany, 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' September - l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Oktober, 2021 P- 314, 907-941 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='18420/infolmatik2021-078, https : //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='18420/informatik2021-078 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Barker, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Polk, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Souppaya, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=': Getting Ready for Post-Quantum Cryp- tography:: Explore Challenges Associated with Adoption and Use of Post- Quantum Cryptographic Algorithms (May 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='6028/ NIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='CSWP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='05262020-draft, https://nvlpubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='gov/nistpubs/CSWP/NIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' cswp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 05262020-draf t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='pdf 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Becker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Knackstedt, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', POppelbulé, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=': Developing maturity models for management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' (3), 213-222 (06 2009) 1 IT 20 von Nethen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Bundesamt für Sicherheit in der Informationstechnik (BSI): Migration zu Post-Quanten-Kryptografie Handlungsempfehlungen des BSI p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 9 (Aug 2020), https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='bsi.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Springer International Publishing, Cham (2021), http://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='com/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='1007/ 978-3-030-50502-8_4, series Title: Applied Innovation and Technology Manage- ment 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Putnam, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Myers, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=': Year 2000 work comes down to the wire 16 96(1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='1109/52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='744575,http://ieeexplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='org/ document/744575/ 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Schultz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=': Managing a y2k project-starting now 15 71 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' https:// doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='1109/52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='676742,http://ieeexplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='org/document/676742/ 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Shor, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' : Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' SIAM Journal on Computing 26 1509 UDct 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='1137/S0097539795293172,http://epubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' siam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='1137/S0097539795293172 (3), 63- (1), 90- (5),l484- PMMP 21 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' TÜV Informationstechnik GmbH: Whitepaper Post-Quantum Security (Nov 2020), https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='tuvit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='de/en/innovations/post-quantum-cryptography/ #c530188 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' utimaco IS GmbH: Post-Quanten-Kryptografie: Sichere Verschlüsselung für das Quanten-Zeitalter (Mar 2018), https : I/www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' infopoint-security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='de/media/ Utimaco_Whitepaper_0uantum-Computing_DE_vfinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='pdf 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Miranskyy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Rjaibi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=': Quantum advantage a d the y2k bug: A comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' IEEE Software 38(2), 80-87 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 1109/MS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='2985321, Conference Name: IEEE Software 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Miranskyy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Rjaibi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Stager, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Gray, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Peck, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=': Making Ex- isting Software Quantum Safe: Lessons Learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='08661 [es] (Oct 2021), http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='org/abs/2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='08661, archiv: 2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='08661 A CAl\\/Il\\/I requirements PMMP m e t s the requirements of the cryptographic agility maturity model de- veloped in [10] As show in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' White the CAMM requirements are intended to measure (not establish) the crypto-agility of Systems, PMMP deliv- ers processes hose resultat fulfill the desired CAMM requirements up to CAMM level 3 (practiced crypto-agility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' We also take a look at CAMM level 4 require- ments (sophisticated erypto-agility) und their relation to PMMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='1 CAl\\/Il\\/I level 1: "possible" R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='0: System knowledge PMMP fulfills This requirement by involving stake- holders of the organization a d compiling a list of Communication partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' By that, the organization understands its context a d is able to evaluate the Impact quantum Computers Wright have on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' PMMP Features a strong Focus on Fisk management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='1: Updateability PMMP fulfills This requirement by providing processes Voith the Goal to Supply the needed resources for necessary Updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For exam- ple, the executive management is educated in cryptographic a d quantum Fisk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Then, the management can provide monetary resourees to finanee the improving process of crypto-agility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To prevent restricting the functionality of Systems, their requirements need to be Clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Then, it an be measured if any functionality is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' PMMP involves reverse engineering processes where needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='2: Extensibility This requirement, in contrast to the requirement Update- ability above, is not about assessing the ability to nun komplement PQC, but actually about acquiring resources needed for the Upgrade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Resources have to be approved to, er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', buy new hardware, a d allow for external Support while migrating cryptography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' PMMP 21 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' TUV Informationsteohnik GmbH: Whitepaper Post-Quantum Security (Nov 20201, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='tuvit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='de/en/innovations/post-quantum-cryptography/ #0530188 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' utimaco IS GmbH: Post-Quanten-Kryptografie: Sichere Verschliisselung fur das Quanten-Zeitalter (Mar 20181, https : I/www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' infopoint-security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='de/media/ Utimaco_Whitepaper_0uantum-Computing_DE_vfinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='pdf 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Miranskyy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Rjaibi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=': Quantum advantage and the y2k bug: A comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' IEEE Software 38(2), 80-87 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 1109/MS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='2985321, conference Name: IEEE Software 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Miranskyy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Rjaibi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Stager, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Gray, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', Peck, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=': Making Ex- isting Software Quantum Safe: Lessons Learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='08661 [cs] (Oct 2021), http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='org/abs/2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='08661, arXiv: 211008661 A CAl\\/IM requirements PMMP meets the requirements of the cryptographic agility maturity model de- veloped in 1101 as shown in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' While the CAMM requirements are intended to measure (not establish) the crypto-agility of systems, PMMP deliv- ers processes whose results fulfill the desired CAMM requirements up to CAMM level 3 (practiced crypto-agility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' We also take a look at CAMM level 4 require- ments (sophisticated crypto-agility) and their relation to PMMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='1 CAl\\/Il\\/I level 1: "possible" R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='0: System knowledge PMMP fulfills this requirement by involving stake- holders of the organization and compiling a list of communication partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' By that, the organization understands its context and is able to evaluate the impact quantum computers might have on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' PMMP features a strong focus on risk management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='1: Updateability PMMP fulfills this requirement by providing processes with the goal to supply the needed resources for necessary updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For exam- ple, the executive management is educated in cryptographic and quantum risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Then, the management can provide monetary resources to finance the improving process of crypto-agility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To prevent restricting the functionality of systems, their requirements need to be clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Then, it can be measured if any functionality is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' PMMP involves reverse engineering processes where needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='2: Extensibility This requirement, in contrast to the requirement Update- ability above, is not about assessing the ability to run implement PQC, but actually about acquiring resources needed for the upgrade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Resources have to be approved to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', buy new hardware, and allow for external support while migrating cryptography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 22 von Nethen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='3: Reversibility As the migration to PQC is managed per application, Voith each application getting migrated in a dedicated project, this requirement is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Also, PMMP involves sing pilot Systems to ensure the Updates work as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='4: Cryptography inventory PMMP has defined processes for This re- quirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' It deines how the inventory of cryptography Can be compiled in an organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In conjunction Voith Fisk management, the level of Security the cryptographic primitives provide a d which level of Security the data handled by the applications where the primitives are used is understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Plus, the process compiles an inventory of Communication partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='2 CA1\\/IM l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' evel 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' upr6p&r6dw R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='0: Cryptographic modularity tems of an organization are upgraded PMMP ensures that applications a d sys- in Groups based on their business process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' ID R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='1: Algorithm S R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='2: Algorithm intersection R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='3: Algo- rithm exclusion PMMP has a strong focus on the context of the organization, including analyzing its Stakeholders und Communication partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In consulta- tion Voith the Communication partners and especially their eapabilíties in the changeover to PQC, PMMP enables a regulated changeover of the algorithmus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The use of standardized algorithm identifiers is suggested for mentioned consul- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In addition, the intersection and exclusion of algorithmus are determined in the reconciliations Voith the Communication partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Furthermore, Fisk-based regulations can ensure that certain algorithmus are excluded, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='4: Opportunistic Security PMMP also deals Voith this issue fror The background of the Communication partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' If no individual coordination Can be made Voith a large number of partners, as is the Case, for example, Voith the visitors to a globally accessible Website, the process relies on opportunistic Security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For this purpose, the Users of the Systems are analyzed beforehand and the best possible procedures are ensured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='3 CAl\\/Il\\/I level 3: "practiced" R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='0: Policies PMMP has a Great Focus on Security Management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For example, the migration process initiales changes to an existing Security policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The Security policy is required to State hat the organization warts in terms of migration to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The policy clarifies which requirements existing and Future applications have to fulfill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Policies are developed in reconciliation Voith the Communication partners, that Wright be required to also migrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 22 von Nethen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='3: Reversibility As the migration to PQC is managed per application, with each application getting migrated in a dedicated project, this requirement is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Also, PMMP involves using pilot systems to ensure the updates work as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='4: Cryptography inventory PMMP has defined processes for this re- quirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' It defines how the inventory of cryptography can be compiled in an organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In conjunction with risk management, the level of security the cryptographic primitives provide and which level of security the data handled by the applications where the primitives are used is understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Plus, the process compiles an inventory of communication partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='2 CA1\\/IM l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' evel 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' upr6p&r6dw R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='0: Cryptographic modularity rems of an organization are upgraded PMMP ensures that applications and sys- in groups based on their business process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' ID R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='1: Algorithm S R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='2: Algorithm intersection R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='3: Algo- rithm exclusion PMMP has a strong focus on the context of the organization, including analyzing its stakeholders and communication partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In consulta- tion with the communication partners and especially their capabilities in the changeover to PQC, PMMP enables a regulated changeover of the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The use of standardized algorithm identifiers is suggested for mentioned consul- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' In addition, the intersection and exclusion of algorithms are determined in the reconciliations with the communication partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Furthermore, risk-based regulations can ensure that certain algorithms are excluded, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='4: Opportunistic security PMMP also deals with this issue from the background of the communication partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' If no individual coordination can be made with a large number of partners, as is the case, for example, with the visitors to a globally accessible website, the process relies on opportunistic security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For this purpose, the users of the systems are analyzed beforehand and the best possible procedures are ensured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='3 CAMM level 3: "practiced" R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content="'8." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='0: Policies PMMP has a great focus on security management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For example, the migration process initiates changes to an existing security policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The security policy is required to state what the organization wants in terms of migration to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The policy clarifies which requirements existing and future applications have to fulfill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Policies are developed in reconciliation with the communication partners, that might be required to also migrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' PMMP 23 R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='1: Performance awareness PMMP takes into Account Security risks und economic risks, such As losing customers that und themselves u sing a device fly capable of performing a Slow post-quantum Secure TLS handshake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' PMMP deines the process of either accepting the risks er stocking up hardware, at least on on Side of the Connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='2: Hardware modularity PMMP fulfills This application-specific require- ment by including a refurbishment process in the migration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For applica- tions, for whieh the Fisk assessment revealed that they need to get migrated and where the organization has the ability to influence the Software a d hardware, making the needed hanges to komplement PQC, the migration proeess ensures that needed hanges are made file migrating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='3: Testing PMMP integrales into existing Management processes like Fisk management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Because the Fisk assessment is not a one-shot process, it is regularly checked whether the Security needs of the organization are fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='4: Enforceability PMMP is drive by the leader of an organization, hat is, the migration process is applied fror top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' A Security policy put in place by the organization°s executive management ensures that the needed techniques, especially resources, are made available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, if resources are lacking, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', educational resources, the migration process recommends co- operating Voith universities and research institutions in the Geld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='5: Security White PMMP is Applied, the Security of the organization is assessed sing external Security audits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To ensure the organization is not vulner- able to attacks, Fisk Management assesses the seeurity of the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='6: Backwards compatibility This requirement is fulfilled by PMMP through its Focus on interoperabílity, which includes berg backward compat- ible Voith oder Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, the process presents different management strategies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', the incremental packet conversion approach (fror [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' With the incremental packet conversion it is ensured, that the different parts of the ap- plication migrated stay compatible Voith each Other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Moreover, the migration process makes se of different transition mechanísms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='7: Transition nıechanisnıs One technique that is used by PMMP is the se of Gateways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This idem is taken fror the REMIP presented in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Therefore, PMMP Features processes to fulfill this requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='8: Effectiveness PMMP makes se of project Management techniques to ensure upgrading cryptographíe primítíves is effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Moreover, PMMP Can PMMP 23 R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='1: Performance awareness PMMP takes into account security risks and economic risks, such as losing customers that find themselves using a device only capable of performing a slow post-quantum secure TLS handshake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' PMMP defines the process of either accepting the risks or stocking up hardware, at least on one side of the connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='2: Hardware modularity PMMP fulfills this application-specific require- ment by including a refurbishment process in the migration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For applica- tions, for which the risk assessment revealed that they need to get migrated and where the organization has the ability to influence the software and hardware, making the needed changes to implement PQC, the migration process ensures that needed changes are made while migrating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='3: Testing PMMP integrates into existing management processes like risk management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Because the risk assessment is not a one-shot process, it is regularly checked whether the security needs of the organization are fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='4: Enforceability PMMP is driven by the leaders of an organization, that is, the migration process is applied from top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=" A security policy put in place by the organization's executive management ensures that the needed techniques, especially resources, are made available." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, if resources are lacking, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', educational resources, the migration process recommends co- operating with universities and research institutions in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='5: Security While PMMP is applied, the security of the organization is assessed using external security audits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' To ensure the organization is not vulner- able to attacks, risk management assesses the security of the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='6: Backwards compatibility This requirement is fulfilled by PMMP through its focus on interoperability, which includes being backward compat- ible with older systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, the process presents different management strategies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=', the incremental packet conversion approach (from 191l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' With the incremental packet conversion it is ensured, that the different parts of the ap- plication migrated stay compatible with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Moreover, the migration process makes use of different transition mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=" '7: Transition mechanisms One technique that is used by PMMP is the use of gateways." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This idea is taken from the REMIP presented in 191 Therefore, PMMP features processes to fulfill this requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='8: Effectiveness PMMP makes use of project management techniques to ensure upgrading cryptographic primitives is effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Moreover, PMMP can 24 von Nethen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' be monitored by the Internal control System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Theo, a lack of knowledge Can be measured und reacted upon, for instanz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, PMMP features an evaluation at the end of every application that is migrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The Goal of this step is to und ways to improve the Overall process, which ensures the effectiveness of the Migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Also, PMMP provides Solutions for Systems that Carnot get upgraded in time ( a d have to be replaced Voith a Brewer application) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='4 CAl\\/Il\\/I level 4: "sophisticated" R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='0: Automation PMMP d e s not include processes for The Automation on crypto-agility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Processes that could be included in the Migration are continuous Integration a d continuous deployment pipelines that could be used to test the applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Then, sing an automated process, the applications eould be auto- matically checked whether they fulfill the requirements for crypto-agility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' If not , they could be automatically upgraded to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='1: Context Independence The techniques presented in PMMP are not linked to a specific context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Rather, weithin the migration process, the eontext of Migration is developed to match the relevant Scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, different application architectures are considered in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Therefore, the migration process Can be used in different contexts a d is context-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='2: Scalability This requirement is not supported by PMMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The migration process has to be adapted to specific organizations a d Carnot be used for every possible organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='3: Real-time PMMP includes processes that help Voith migrating in a g i e n time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Moreover, there are techniques defined that strueture the Migration timely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' But, a Migration in real-time is not possible Voith PMMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Techniques that could allow a real-time migration are not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='4: Interoperability Interoperability between various Systems is considered in PMMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For example, the se of gateways is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This allows applica- tions that Carnot get upgraded to stay connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Moreover, the process has a strong focus on the context of the organization whieh ineludes its Stakeholders (for example Users of a Website).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='5 Results In the above evaluation against the requirements of CAMM, it is argued that PMMP includes methods to Upgrade applications up to level 3 "practiced" of CAMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The requirements of level 4 cc sophistieated" are not all satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' 24 von Nethen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' be monitored by the internal control system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Then, a lack of knowledge can be measured and reacted upon, for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, PMMP features an evaluation at the end of every application that is migrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The goal of this step is to find ways to improve the overall process, which ensures the effectiveness of the migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Also, PMMP provides solutions for systems that cannot get upgraded in time (and have to be replaced with a newer application) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='4 CAMM level 4: "sophisticated" R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='0: Automation PMMP does not include processes for the automation of crypto-agility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Processes that could be included in the migration are continuous integration and continuous deployment pipelines that could be used to test the applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Then, using an automated process, the applications could be auto- matically checked whether they fulfill the requirements for crypto-agility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' If not , they could be automatically upgraded to PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='1: Context independence The techniques presented in PMMP are not linked to a specific context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Rather, within the migration process, the context of migration is developed to match the relevant scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Additionally, different application architectures are considered in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Therefore, the migration process can be used in different contexts and is context-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='2: Scalability This requirement is not supported by PMMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The migration process has to be adapted to specific organizations and cannot be used for every possible organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='3: Real-time PMMP includes processes that help with migrating in a given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Moreover, there are techniques defined that structure the migration timely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' But, a migration in real-time is not possible with PMMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Techniques that could allow a real-time migration are not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='4: Interoperability Interoperability between various systems is considered in PMMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' For example, the use of gateways is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' This allows applica- tions that cannot get upgraded to stay connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' Moreover, the process has a strong focus on the context of the organization which includes its stakeholders for example users of a website).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content='5 Results In the above evaluation against the requirements of CAMM, it is argued that PMMP includes methods to upgrade applications up to level 3 "practiced" of CAMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} +page_content=' The requirements of level 4 cc sophisticated" are not all satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE3T4oBgHgl3EQfYwqL/content/2301.04491v1.pdf'} diff --git a/NtFLT4oBgHgl3EQfOS8u/content/tmp_files/2301.12023v1.pdf.txt b/NtFLT4oBgHgl3EQfOS8u/content/tmp_files/2301.12023v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..84c1b3510a2c669a70f79c02b4ecc65867a6d5e2 --- /dev/null +++ b/NtFLT4oBgHgl3EQfOS8u/content/tmp_files/2301.12023v1.pdf.txt @@ -0,0 +1,1533 @@ +META TEMPORAL POINT PROCESSES +Wonho Bae +University of British Columbia & Borealis AI +whbae@cs.ubc.ca +Mohamed Osama Ahmed +Borealis AI +mohamed.o.ahmed@borealisai.com +Frederick Tung +Borealis AI +frederick.tung@borealisai.com +Gabriel L. Oliveira +Borealis AI +gabriel.oliveira@borealisai.com +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